70:
1155:
of the octonions. A general element in SO(8) can be described as the product of 7 left-multiplications, 7 right-multiplications and also 7 bimultiplications by unit octonions (a bimultiplication being the composition of a left-multiplication and a right-multiplication by the same octonion and is
2718:
1163:
It can be shown that an element of SO(8) can be constructed with bimultiplications, by first showing that pairs of reflections through the origin in 8-dimensional space correspond to pairs of bimultiplications by unit octonions. The
1699:
1775:
2588:
1623:
1413:
1491:
1577:
1040:
vector representation, of Spin(8) are all eight-dimensional (for all other spin groups the spinor representation is either smaller or larger than the vector representation). The triality
1877:
2158:
2122:
2084:
2046:
1941:
1813:
1360:
2194:
1909:
1451:
1214:
2555:
2504:
2453:
2402:
2351:
2300:
2240:
515:
490:
453:
2214:
1833:
1511:
1290:
1252:
882:
2008:
1976:
1322:
817:
1625:
be the corresponding products of left and right multiplications by the conjugates (i.e., the multiplicative inverses) of the same unit octonions, so
375:
2778:
1628:
1704:
325:
31:
2713:{\displaystyle {\begin{pmatrix}2&-1&-1&-1\\-1&2&0&0\\-1&0&2&0\\-1&0&0&2\end{pmatrix}}}
810:
320:
2860:
2830:
2809:
1005:
1010:
1000:
990:
1582:
736:
1168:
automorphism of Spin(8) described below provides similar constructions with left multiplications and right multiplications.
995:
803:
1365:
1456:
2801:
1116:
Sometimes Spin(8) appears naturally in an "enlarged" form, as the automorphism group of Spin(8), which breaks up as a
420:
234:
2852:
1516:
1037:
2837:
1838:
1096:
1045:
839:
618:
352:
229:
117:
2131:
2089:
2051:
2013:
1914:
1780:
1327:
2167:
1882:
1815:
is an isotopy. As a result of the non-associativity of the octonions, the only other orthogonal isotopy for
1422:
1179:
768:
558:
642:
2510:
2459:
2408:
2357:
2306:
2255:
1078:
1033:
582:
570:
188:
122:
2219:
2878:
157:
52:
498:
473:
436:
2125:
1157:
1117:
142:
114:
2856:
2826:
2805:
2774:
1152:
924:
889:
713:
547:
390:
284:
2199:
1818:
1496:
1257:
1219:
2818:
2739:
2734:
1416:
1015:
979:
885:
861:
847:
698:
690:
682:
674:
666:
654:
594:
534:
524:
366:
308:
183:
152:
34:
1981:
1949:
1295:
2844:
1049:
905:
843:
782:
775:
761:
718:
606:
529:
359:
273:
213:
93:
983:
789:
725:
415:
395:
332:
297:
218:
208:
193:
178:
132:
109:
41:
2872:
2579:
2572:
1056:
that permutes these three representations. The automorphism group acts on the center
708:
630:
464:
337:
203:
1140:
1041:
896:
563:
262:
251:
198:
173:
168:
127:
98:
61:
2768:
2793:
2246:
831:
30:
2568:
2561:
2161:
909:
892:
730:
458:
2729:
1513:
can be described as the product of bimultiplications of unit octonions, say
1136:
551:
69:
1165:
1026:
1022:
973:
88:
430:
344:
17:
1946:
Multiplicative inverses of octonions are two-sided, which means that
1694:{\displaystyle \alpha =L_{\overline {u_{1}}}...L_{\overline {u_{n}}}}
1148:
1030:
1770:{\displaystyle \beta =R_{\overline {u_{1}}}...R_{\overline {u_{n}}}}
1151:. However the relationship is more complicated, partly due to the
1144:
29:
1106:. The triality symmetry acts again on the further quotient SO(8)/
1879:. As the set of orthogonal isotopies produce a 2-to-1 cover of
1139:, analogously to how elements of SO(2) can be described with
1095:, breaking this symmetry and obtaining SO(8), the remaining
2851:, Cambridge Studies in Advanced Mathematics, vol. 50,
1025:. This gives rise to peculiar feature of Spin(8) known as
2048:
can be permuted cyclically to give two further isotopies
1021:
under the Dynkin classification), possesses a three-fold
1618:{\displaystyle \alpha ,\beta \in \operatorname {SO(8)} }
2597:
2823:
The algebraic theory of spinors and
Clifford algebras
2591:
2513:
2462:
2411:
2360:
2309:
2258:
2222:
2202:
2170:
2134:
2092:
2054:
2016:
1984:
1952:
1917:
1885:
1841:
1821:
1783:
1707:
1631:
1585:
1519:
1499:
1459:
1425:
1368:
1330:
1298:
1260:
1222:
1182:
1088:≅GL(2,2)). When one quotients Spin(8) by one central
864:
501:
476:
439:
2575:
has 4! × 8 = 192 elements.
2160:. This "triality" automorphism is exceptional among
1156:unambiguously defined due to octonions obeying the
2767:John H. Conway; Derek A. Smith (23 January 2003).
2712:
2549:
2498:
2447:
2396:
2345:
2294:
2234:
2208:
2188:
2152:
2116:
2078:
2040:
2002:
1970:
1935:
1903:
1871:
1827:
1807:
1769:
1693:
1617:
1571:
1505:
1485:
1453:, the isotopy is called an orthogonal isotopy. If
1445:
1408:{\displaystyle x^{\alpha }y^{\beta }z^{\gamma }=1}
1407:
1354:
1316:
1284:
1246:
1208:
876:
509:
484:
447:
2825:, Collected works, vol. 2, Springer-Verlag,
2762:
2760:
1486:{\displaystyle \gamma \in \operatorname {SO(8)} }
1070:(which also has automorphism group isomorphic to
27:Rotation group in 8-dimensional Euclidean space
1572:{\displaystyle \gamma =B_{u_{1}}...B_{u_{n}}}
1254:, it can be shown that this is equivalent to
1135:Elements of SO(8) can be described with unit
934:, the diagonal matrices {±I} (as for all SO(2
811:
8:
1029:. Related to this is the fact that the two
2849:Clifford algebras and the classical groups
1872:{\displaystyle (-\alpha ,-\beta ,\gamma )}
1081:over the finite field with two elements,
818:
804:
256:
82:
47:
2592:
2590:
2512:
2461:
2410:
2359:
2308:
2257:
2242:are only uniquely determined up to sign.
2221:
2201:
2169:
2133:
2091:
2053:
2015:
1983:
1951:
1916:
1884:
1840:
1820:
1782:
1755:
1749:
1724:
1718:
1706:
1679:
1673:
1648:
1642:
1630:
1598:
1584:
1561:
1556:
1535:
1530:
1518:
1498:
1466:
1458:
1426:
1424:
1419:. If the three maps of an isotopy are in
1393:
1383:
1373:
1367:
1329:
1297:
1259:
1221:
1202:
1201:
1181:
863:
503:
502:
500:
478:
477:
475:
441:
440:
438:
2153:{\displaystyle \operatorname {Spin} (8)}
2117:{\displaystyle (\gamma ,\alpha ,\beta )}
2079:{\displaystyle (\beta ,\gamma ,\alpha )}
2041:{\displaystyle (\alpha ,\beta ,\gamma )}
1936:{\displaystyle \operatorname {Spin} (8)}
1808:{\displaystyle (\alpha ,\beta ,\gamma )}
1355:{\displaystyle (\alpha ,\beta ,\gamma )}
2756:
2164:. There is no triality automorphism of
846:. It could be either a real or complex
374:
140:
50:
2189:{\displaystyle \operatorname {SO} (8)}
1904:{\displaystyle \operatorname {SO} (8)}
1446:{\displaystyle \operatorname {SO(8)} }
1048:of Spin(8) which is isomorphic to the
858:Like all special orthogonal groups of
376:Classification of finite simple groups
1362:that preserve this identity, so that
1209:{\displaystyle x,y,z\in \mathbb {O} }
942:≥ 4), while the center of Spin(8) is
7:
1324:without ambiguity. A triple of maps
1077:which may also be considered as the
2800:, Chicago Lectures in Mathematics,
2798:Lectures on exceptional Lie groups
2010:. This means that a given isotopy
1777:. A simple calculation shows that
1602:
1599:
1470:
1467:
1430:
1427:
25:
2836:(originally published in 1954 by
2550:{\displaystyle (0,0,\pm 1,\pm 1)}
2499:{\displaystyle (0,\pm 1,0,\pm 1)}
2448:{\displaystyle (0,\pm 1,\pm 1,0)}
2397:{\displaystyle (\pm 1,0,0,\pm 1)}
2346:{\displaystyle (\pm 1,0,\pm 1,0)}
2295:{\displaystyle (\pm 1,\pm 1,0,0)}
1008:
1003:
998:
993:
988:
68:
2544:
2514:
2493:
2463:
2442:
2412:
2391:
2361:
2340:
2310:
2289:
2259:
2235:{\displaystyle \alpha ,\beta }
2183:
2177:
2147:
2141:
2111:
2093:
2073:
2055:
2035:
2017:
1930:
1924:
1898:
1892:
1866:
1842:
1802:
1784:
1611:
1605:
1479:
1473:
1439:
1433:
1349:
1331:
1273:
1264:
1232:
1223:
737:Infinite dimensional Lie group
1:
2770:On Quaternions and Octonions
1761:
1730:
1685:
1654:
850:of rank 4 and dimension 28.
842:acting on eight-dimensional
510:{\displaystyle \mathbb {Z} }
485:{\displaystyle \mathbb {Z} }
448:{\displaystyle \mathbb {Z} }
2802:University of Chicago Press
2124:. This produces an order 3
1493:, then following the above
1120:: Aut(Spin(8)) ≅ PSO (8) ⋊
235:List of group theory topics
2895:
2853:Cambridge University Press
978:SO(8) is unique among the
971:
2838:Columbia University Press
2773:. Taylor & Francis.
1097:outer automorphism group
1046:outer automorphism group
1044:of Spin(8) lives in the
840:special orthogonal group
353:Elementary abelian group
230:Glossary of group theory
2216:the corresponding maps
2209:{\displaystyle \gamma }
1911:, they must in fact be
1828:{\displaystyle \gamma }
1506:{\displaystyle \gamma }
1285:{\displaystyle x(yz)=1}
1247:{\displaystyle (xy)z=1}
2714:
2551:
2500:
2449:
2398:
2347:
2296:
2236:
2210:
2190:
2154:
2118:
2080:
2042:
2004:
1972:
1937:
1905:
1873:
1829:
1809:
1771:
1695:
1619:
1573:
1507:
1487:
1447:
1409:
1356:
1318:
1286:
1248:
1210:
1172:Octonions and triality
1147:can be described with
878:
877:{\displaystyle n>2}
769:Linear algebraic group
511:
486:
449:
45:
2715:
2552:
2501:
2450:
2399:
2348:
2297:
2237:
2211:
2191:
2155:
2119:
2081:
2043:
2005:
2003:{\displaystyle yzx=1}
1973:
1971:{\displaystyle xyz=1}
1938:
1906:
1874:
1830:
1810:
1772:
1696:
1620:
1574:
1508:
1488:
1448:
1410:
1357:
1319:
1317:{\displaystyle xyz=1}
1287:
1249:
1211:
879:
512:
487:
450:
33:
2589:
2511:
2460:
2409:
2358:
2307:
2256:
2220:
2200:
2168:
2132:
2090:
2052:
2014:
1982:
1950:
1915:
1883:
1839:
1819:
1781:
1705:
1629:
1583:
1517:
1497:
1457:
1423:
1366:
1328:
1296:
1258:
1220:
1180:
1141:unit complex numbers
1079:general linear group
862:
499:
474:
437:
143:Group homomorphisms
53:Algebraic structure
2710:
2704:
2547:
2496:
2445:
2394:
2343:
2292:
2232:
2206:
2186:
2150:
2126:outer automorphism
2114:
2076:
2038:
2000:
1968:
1933:
1901:
1869:
1825:
1805:
1767:
1691:
1615:
1569:
1503:
1483:
1443:
1405:
1352:
1314:
1282:
1244:
1206:
1158:Moufang identities
1118:semidirect product
956:(as for all Spin(4
874:
619:Special orthogonal
507:
482:
445:
326:Lagrange's theorem
46:
2819:Chevalley, Claude
2780:978-1-56881-134-5
2196:, as for a given
1978:is equivalent to
1764:
1733:
1688:
1657:
1153:non-associativity
1036:, as well as the
980:simple Lie groups
890:fundamental group
828:
827:
403:
402:
285:Alternating group
242:
241:
16:(Redirected from
2886:
2865:
2845:Porteous, Ian R.
2835:
2814:
2785:
2784:
2764:
2735:Clifford algebra
2719:
2717:
2716:
2711:
2709:
2708:
2556:
2554:
2553:
2548:
2505:
2503:
2502:
2497:
2454:
2452:
2451:
2446:
2403:
2401:
2400:
2395:
2352:
2350:
2349:
2344:
2301:
2299:
2298:
2293:
2241:
2239:
2238:
2233:
2215:
2213:
2212:
2207:
2195:
2193:
2192:
2187:
2159:
2157:
2156:
2151:
2123:
2121:
2120:
2115:
2085:
2083:
2082:
2077:
2047:
2045:
2044:
2039:
2009:
2007:
2006:
2001:
1977:
1975:
1974:
1969:
1942:
1940:
1939:
1934:
1910:
1908:
1907:
1902:
1878:
1876:
1875:
1870:
1834:
1832:
1831:
1826:
1814:
1812:
1811:
1806:
1776:
1774:
1773:
1768:
1766:
1765:
1760:
1759:
1750:
1735:
1734:
1729:
1728:
1719:
1700:
1698:
1697:
1692:
1690:
1689:
1684:
1683:
1674:
1659:
1658:
1653:
1652:
1643:
1624:
1622:
1621:
1616:
1614:
1578:
1576:
1575:
1570:
1568:
1567:
1566:
1565:
1542:
1541:
1540:
1539:
1512:
1510:
1509:
1504:
1492:
1490:
1489:
1484:
1482:
1452:
1450:
1449:
1444:
1442:
1414:
1412:
1411:
1406:
1398:
1397:
1388:
1387:
1378:
1377:
1361:
1359:
1358:
1353:
1323:
1321:
1320:
1315:
1291:
1289:
1288:
1283:
1253:
1251:
1250:
1245:
1215:
1213:
1212:
1207:
1205:
1149:unit quaternions
1143:and elements of
1013:
1012:
1011:
1007:
1006:
1002:
1001:
997:
996:
992:
991:
908:of SO(8) is the
886:simply connected
883:
881:
880:
875:
848:simple Lie group
820:
813:
806:
762:Algebraic groups
535:Hyperbolic group
525:Arithmetic group
516:
514:
513:
508:
506:
491:
489:
488:
483:
481:
454:
452:
451:
446:
444:
367:Schur multiplier
321:Cauchy's theorem
309:Quaternion group
257:
83:
72:
59:
48:
21:
2894:
2893:
2889:
2888:
2887:
2885:
2884:
2883:
2869:
2868:
2863:
2843:
2833:
2817:
2812:
2792:
2789:
2788:
2781:
2766:
2765:
2758:
2753:
2745:
2726:
2703:
2702:
2697:
2692:
2687:
2678:
2677:
2672:
2667:
2662:
2653:
2652:
2647:
2642:
2637:
2628:
2627:
2619:
2611:
2603:
2593:
2587:
2586:
2583:
2565:
2509:
2508:
2458:
2457:
2407:
2406:
2356:
2355:
2305:
2304:
2254:
2253:
2250:
2218:
2217:
2198:
2197:
2166:
2165:
2130:
2129:
2088:
2087:
2050:
2049:
2012:
2011:
1980:
1979:
1948:
1947:
1913:
1912:
1881:
1880:
1837:
1836:
1817:
1816:
1779:
1778:
1751:
1745:
1720:
1714:
1703:
1702:
1675:
1669:
1644:
1638:
1627:
1626:
1581:
1580:
1557:
1552:
1531:
1526:
1515:
1514:
1495:
1494:
1455:
1454:
1421:
1420:
1389:
1379:
1369:
1364:
1363:
1326:
1325:
1294:
1293:
1292:, meaning that
1256:
1255:
1218:
1217:
1178:
1177:
1174:
1133:
1126:
1112:
1105:
1094:
1087:
1076:
1069:
1062:
1055:
1050:symmetric group
1034:representations
1019:
1009:
1004:
999:
994:
989:
987:
976:
970:
955:
948:
933:
921:
906:universal cover
902:
884:, SO(8) is not
860:
859:
856:
844:Euclidean space
824:
795:
794:
783:Abelian variety
776:Reductive group
764:
754:
753:
752:
751:
702:
694:
686:
678:
670:
643:Special unitary
554:
540:
539:
521:
520:
497:
496:
472:
471:
435:
434:
426:
425:
416:Discrete groups
405:
404:
360:Frobenius group
305:
292:
281:
274:Symmetric group
270:
254:
244:
243:
94:Normal subgroup
80:
60:
51:
38:
28:
23:
22:
15:
12:
11:
5:
2892:
2890:
2882:
2881:
2871:
2870:
2867:
2866:
2861:
2841:
2831:
2815:
2810:
2787:
2786:
2779:
2755:
2754:
2752:
2749:
2748:
2747:
2743:
2737:
2732:
2725:
2722:
2721:
2720:
2707:
2701:
2698:
2696:
2693:
2691:
2688:
2686:
2683:
2680:
2679:
2676:
2673:
2671:
2668:
2666:
2663:
2661:
2658:
2655:
2654:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2630:
2629:
2626:
2623:
2620:
2618:
2615:
2612:
2610:
2607:
2604:
2602:
2599:
2598:
2596:
2582:
2577:
2564:
2559:
2558:
2557:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2506:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2455:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2404:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2353:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2302:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2249:
2244:
2231:
2228:
2225:
2205:
2185:
2182:
2179:
2176:
2173:
2149:
2146:
2143:
2140:
2137:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
1999:
1996:
1993:
1990:
1987:
1967:
1964:
1961:
1958:
1955:
1932:
1929:
1926:
1923:
1920:
1900:
1897:
1894:
1891:
1888:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1824:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1763:
1758:
1754:
1748:
1744:
1741:
1738:
1732:
1727:
1723:
1717:
1713:
1710:
1687:
1682:
1678:
1672:
1668:
1665:
1662:
1656:
1651:
1647:
1641:
1637:
1634:
1613:
1610:
1607:
1604:
1601:
1597:
1594:
1591:
1588:
1564:
1560:
1555:
1551:
1548:
1545:
1538:
1534:
1529:
1525:
1522:
1502:
1481:
1478:
1475:
1472:
1469:
1465:
1462:
1441:
1438:
1435:
1432:
1429:
1404:
1401:
1396:
1392:
1386:
1382:
1376:
1372:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1313:
1310:
1307:
1304:
1301:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1204:
1200:
1197:
1194:
1191:
1188:
1185:
1173:
1170:
1132:
1131:Unit octonions
1129:
1124:
1110:
1103:
1092:
1085:
1074:
1067:
1060:
1053:
1017:
984:Dynkin diagram
972:Main article:
969:
966:
953:
946:
931:
920:
917:
900:
873:
870:
867:
855:
852:
826:
825:
823:
822:
815:
808:
800:
797:
796:
793:
792:
790:Elliptic curve
786:
785:
779:
778:
772:
771:
765:
760:
759:
756:
755:
750:
749:
746:
743:
739:
735:
734:
733:
728:
726:Diffeomorphism
722:
721:
716:
711:
705:
704:
700:
696:
692:
688:
684:
680:
676:
672:
668:
663:
662:
651:
650:
639:
638:
627:
626:
615:
614:
603:
602:
591:
590:
583:Special linear
579:
578:
571:General linear
567:
566:
561:
555:
546:
545:
542:
541:
538:
537:
532:
527:
519:
518:
505:
493:
480:
467:
465:Modular groups
463:
462:
461:
456:
443:
427:
424:
423:
418:
412:
411:
410:
407:
406:
401:
400:
399:
398:
393:
388:
385:
379:
378:
372:
371:
370:
369:
363:
362:
356:
355:
350:
341:
340:
338:Hall's theorem
335:
333:Sylow theorems
329:
328:
323:
315:
314:
313:
312:
306:
301:
298:Dihedral group
294:
293:
288:
282:
277:
271:
266:
255:
250:
249:
246:
245:
240:
239:
238:
237:
232:
224:
223:
222:
221:
216:
211:
206:
201:
196:
191:
189:multiplicative
186:
181:
176:
171:
163:
162:
161:
160:
155:
147:
146:
138:
137:
136:
135:
133:Wreath product
130:
125:
120:
118:direct product
112:
110:Quotient group
104:
103:
102:
101:
96:
91:
81:
78:
77:
74:
73:
65:
64:
42:Dynkin diagram
36:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2891:
2880:
2877:
2876:
2874:
2864:
2862:0-521-55177-3
2858:
2854:
2850:
2846:
2842:
2839:
2834:
2832:3-540-57063-2
2828:
2824:
2820:
2816:
2813:
2811:0-226-00526-7
2807:
2803:
2799:
2795:
2791:
2790:
2782:
2776:
2772:
2771:
2763:
2761:
2757:
2750:
2746:
2742:
2738:
2736:
2733:
2731:
2728:
2727:
2723:
2705:
2699:
2694:
2689:
2684:
2681:
2674:
2669:
2664:
2659:
2656:
2649:
2644:
2639:
2634:
2631:
2624:
2621:
2616:
2613:
2608:
2605:
2600:
2594:
2585:
2584:
2581:
2580:Cartan matrix
2578:
2576:
2574:
2573:Coxeter group
2570:
2563:
2560:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2507:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2456:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2405:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2354:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2303:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2252:
2251:
2248:
2245:
2243:
2229:
2226:
2223:
2203:
2180:
2174:
2171:
2163:
2144:
2138:
2135:
2127:
2108:
2105:
2102:
2099:
2096:
2070:
2067:
2064:
2061:
2058:
2032:
2029:
2026:
2023:
2020:
1997:
1994:
1991:
1988:
1985:
1965:
1962:
1959:
1956:
1953:
1944:
1927:
1921:
1918:
1895:
1889:
1886:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1822:
1799:
1796:
1793:
1790:
1787:
1756:
1752:
1746:
1742:
1739:
1736:
1725:
1721:
1715:
1711:
1708:
1680:
1676:
1670:
1666:
1663:
1660:
1649:
1645:
1639:
1635:
1632:
1608:
1595:
1592:
1589:
1586:
1562:
1558:
1553:
1549:
1546:
1543:
1536:
1532:
1527:
1523:
1520:
1500:
1476:
1463:
1460:
1436:
1418:
1415:is called an
1402:
1399:
1394:
1390:
1384:
1380:
1374:
1370:
1346:
1343:
1340:
1337:
1334:
1311:
1308:
1305:
1302:
1299:
1279:
1276:
1270:
1267:
1261:
1241:
1238:
1235:
1229:
1226:
1198:
1195:
1192:
1189:
1186:
1183:
1171:
1169:
1167:
1161:
1159:
1154:
1150:
1146:
1142:
1138:
1130:
1128:
1123:
1119:
1114:
1109:
1102:
1098:
1091:
1084:
1080:
1073:
1066:
1059:
1051:
1047:
1043:
1039:
1035:
1032:
1028:
1024:
1020:
985:
981:
975:
967:
965:
963:
959:
952:
945:
941:
937:
930:
926:
918:
916:
914:
911:
907:
903:
899:
894:
891:
887:
871:
868:
865:
853:
851:
849:
845:
841:
837:
833:
821:
816:
814:
809:
807:
802:
801:
799:
798:
791:
788:
787:
784:
781:
780:
777:
774:
773:
770:
767:
766:
763:
758:
757:
747:
744:
741:
740:
738:
732:
729:
727:
724:
723:
720:
717:
715:
712:
710:
707:
706:
703:
697:
695:
689:
687:
681:
679:
673:
671:
665:
664:
660:
656:
653:
652:
648:
644:
641:
640:
636:
632:
629:
628:
624:
620:
617:
616:
612:
608:
605:
604:
600:
596:
593:
592:
588:
584:
581:
580:
576:
572:
569:
568:
565:
562:
560:
557:
556:
553:
549:
544:
543:
536:
533:
531:
528:
526:
523:
522:
494:
469:
468:
466:
460:
457:
432:
429:
428:
422:
419:
417:
414:
413:
409:
408:
397:
394:
392:
389:
386:
383:
382:
381:
380:
377:
373:
368:
365:
364:
361:
358:
357:
354:
351:
349:
347:
343:
342:
339:
336:
334:
331:
330:
327:
324:
322:
319:
318:
317:
316:
310:
307:
304:
299:
296:
295:
291:
286:
283:
280:
275:
272:
269:
264:
261:
260:
259:
258:
253:
252:Finite groups
248:
247:
236:
233:
231:
228:
227:
226:
225:
220:
217:
215:
212:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
185:
182:
180:
177:
175:
172:
170:
167:
166:
165:
164:
159:
156:
154:
151:
150:
149:
148:
145:
144:
139:
134:
131:
129:
126:
124:
121:
119:
116:
113:
111:
108:
107:
106:
105:
100:
97:
95:
92:
90:
87:
86:
85:
84:
79:Basic notions
76:
75:
71:
67:
66:
63:
58:
54:
49:
43:
39:
32:
19:
2848:
2822:
2797:
2769:
2740:
2566:
1945:
1175:
1162:
1134:
1121:
1115:
1107:
1100:
1089:
1082:
1071:
1064:
1057:
1042:automorphism
982:in that its
977:
961:
957:
950:
943:
939:
935:
928:
927:of SO(8) is
922:
912:
897:
857:
835:
829:
658:
646:
634:
622:
610:
598:
586:
574:
345:
302:
289:
278:
267:
263:Cyclic group
141:
128:Free product
99:Group action
62:Group theory
57:Group theory
56:
2794:Adams, J.F.
2247:Root system
2162:spin groups
1038:fundamental
888:, having a
832:mathematics
548:Topological
387:alternating
2879:Lie groups
2751:References
2562:Weyl group
910:spin group
893:isomorphic
655:Symplectic
595:Orthogonal
552:Lie groups
459:Free group
184:continuous
123:Direct sum
2730:Octonions
2682:−
2657:−
2632:−
2622:−
2614:−
2606:−
2539:±
2530:±
2488:±
2473:±
2431:±
2422:±
2386:±
2365:±
2329:±
2314:±
2272:±
2263:±
2230:β
2224:α
2204:γ
2175:
2139:
2109:β
2103:α
2097:γ
2071:α
2065:γ
2059:β
2033:γ
2027:β
2021:α
1922:
1890:
1864:γ
1858:β
1855:−
1849:α
1846:−
1823:γ
1800:γ
1794:β
1788:α
1762:¯
1731:¯
1709:β
1686:¯
1655:¯
1633:α
1596:∈
1593:β
1587:α
1521:γ
1501:γ
1464:∈
1461:γ
1395:γ
1385:β
1375:α
1347:γ
1341:β
1335:α
1199:∈
1137:octonions
719:Conformal
607:Euclidean
214:nilpotent
2873:Category
2847:(1995),
2821:(1997),
2796:(1996),
2724:See also
1166:triality
1099:is only
1027:triality
1023:symmetry
974:Triality
968:Triality
938:) with 2
714:Poincaré
559:Solenoid
431:Integers
421:Lattices
396:sporadic
391:Lie type
219:solvable
209:dihedral
194:additive
179:infinite
89:Subgroup
44:of SO(8)
1417:isotopy
913:Spin(8)
854:Spin(8)
838:is the
709:Lorentz
631:Unitary
530:Lattice
470:PSL(2,
204:abelian
115:(Semi-)
18:Spin(8)
2859:
2829:
2808:
2777:
1579:. Let
1031:spinor
964:≥ 4).
949:×
925:center
919:Center
904:. The
564:Circle
495:SL(2,
384:cyclic
348:-group
199:cyclic
174:finite
169:simple
153:kernel
1145:SO(4)
836:SO(8)
748:Sp(∞)
745:SU(∞)
158:image
2857:ISBN
2827:ISBN
2806:ISBN
2775:ISBN
2569:Weyl
2567:Its
2136:Spin
2086:and
1919:Spin
1216:and
960:), 4
923:The
869:>
742:O(∞)
731:Loop
550:and
2128:of
1835:is
1176:If
1160:).
895:to
830:In
657:Sp(
645:SU(
621:SO(
585:SL(
573:GL(
2875::
2855:,
2804:,
2759:^
2172:SO
1943:.
1887:SO
1701:,
1127:.
1113:.
1063:x
986:,
915:.
834:,
633:U(
609:E(
597:O(
55:→
40:,
2840:)
2783:.
2744:2
2741:G
2706:)
2700:2
2695:0
2690:0
2685:1
2675:0
2670:2
2665:0
2660:1
2650:0
2645:0
2640:2
2635:1
2625:1
2617:1
2609:1
2601:2
2595:(
2571:/
2545:)
2542:1
2536:,
2533:1
2527:,
2524:0
2521:,
2518:0
2515:(
2494:)
2491:1
2485:,
2482:0
2479:,
2476:1
2470:,
2467:0
2464:(
2443:)
2440:0
2437:,
2434:1
2428:,
2425:1
2419:,
2416:0
2413:(
2392:)
2389:1
2383:,
2380:0
2377:,
2374:0
2371:,
2368:1
2362:(
2341:)
2338:0
2335:,
2332:1
2326:,
2323:0
2320:,
2317:1
2311:(
2290:)
2287:0
2284:,
2281:0
2278:,
2275:1
2269:,
2266:1
2260:(
2227:,
2184:)
2181:8
2178:(
2148:)
2145:8
2142:(
2112:)
2106:,
2100:,
2094:(
2074:)
2068:,
2062:,
2056:(
2036:)
2030:,
2024:,
2018:(
1998:1
1995:=
1992:x
1989:z
1986:y
1966:1
1963:=
1960:z
1957:y
1954:x
1931:)
1928:8
1925:(
1899:)
1896:8
1893:(
1867:)
1861:,
1852:,
1843:(
1803:)
1797:,
1791:,
1785:(
1757:n
1753:u
1747:R
1743:.
1740:.
1737:.
1726:1
1722:u
1716:R
1712:=
1681:n
1677:u
1671:L
1667:.
1664:.
1661:.
1650:1
1646:u
1640:L
1636:=
1612:)
1609:8
1606:(
1603:O
1600:S
1590:,
1563:n
1559:u
1554:B
1550:.
1547:.
1544:.
1537:1
1533:u
1528:B
1524:=
1480:)
1477:8
1474:(
1471:O
1468:S
1440:)
1437:8
1434:(
1431:O
1428:S
1403:1
1400:=
1391:z
1381:y
1371:x
1350:)
1344:,
1338:,
1332:(
1312:1
1309:=
1306:z
1303:y
1300:x
1280:1
1277:=
1274:)
1271:z
1268:y
1265:(
1262:x
1242:1
1239:=
1236:z
1233:)
1230:y
1227:x
1224:(
1203:O
1196:z
1193:,
1190:y
1187:,
1184:x
1125:3
1122:S
1111:2
1108:Z
1104:2
1101:Z
1093:2
1090:Z
1086:3
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1075:3
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1068:2
1065:Z
1061:2
1058:Z
1054:3
1052:S
1018:4
1016:D
1014:(
962:n
958:n
954:2
951:Z
947:2
944:Z
940:n
936:n
932:2
929:Z
901:2
898:Z
872:2
866:n
819:e
812:t
805:v
701:8
699:E
693:7
691:E
685:6
683:E
677:4
675:F
669:2
667:G
661:)
659:n
649:)
647:n
637:)
635:n
625:)
623:n
613:)
611:n
601:)
599:n
589:)
587:n
577:)
575:n
517:)
504:Z
492:)
479:Z
455:)
442:Z
433:(
346:p
311:Q
303:n
300:D
290:n
287:A
279:n
276:S
268:n
265:Z
37:4
35:D
20:)
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