59:
1144:
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
2707:
1152:
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
1688:
1764:
2577:
1612:
1402:
1480:
1566:
1029:
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
1866:
2147:
2111:
2073:
2035:
1930:
1802:
1349:
2183:
1898:
1440:
1203:
2544:
2493:
2442:
2391:
2340:
2289:
2229:
504:
479:
442:
2203:
1822:
1500:
1279:
1241:
871:
1997:
1965:
1311:
806:
1614:
be the corresponding products of left and right multiplications by the conjugates (i.e., the multiplicative inverses) of the same unit octonions, so
364:
2767:
1617:
1693:
314:
20:
2702:{\displaystyle {\begin{pmatrix}2&-1&-1&-1\\-1&2&0&0\\-1&0&2&0\\-1&0&0&2\end{pmatrix}}}
799:
309:
2849:
2819:
2798:
994:
999:
989:
979:
1571:
725:
1157:
automorphism of Spin(8) described below provides similar constructions with left multiplications and right multiplications.
984:
792:
1354:
1445:
2790:
1105:
Sometimes Spin(8) appears naturally in an "enlarged" form, as the automorphism group of Spin(8), which breaks up as a
409:
223:
2841:
1505:
1026:
2826:
1827:
1085:
1034:
828:
607:
341:
218:
106:
2120:
2078:
2040:
2002:
1903:
1769:
1316:
2156:
1871:
1804:
is an isotopy. As a result of the non-associativity of the octonions, the only other orthogonal isotopy for
1411:
1168:
757:
547:
631:
2499:
2448:
2397:
2346:
2295:
2244:
1067:
1022:
571:
559:
177:
111:
2208:
2867:
146:
41:
487:
462:
425:
2114:
1146:
1106:
131:
103:
2845:
2815:
2794:
2763:
1141:
913:
878:
702:
536:
379:
273:
2188:
1807:
1485:
1246:
1208:
2807:
2728:
2723:
1405:
1004:
968:
874:
850:
836:
687:
679:
671:
663:
655:
643:
583:
523:
513:
355:
297:
172:
141:
23:
1970:
1938:
1284:
2833:
1038:
894:
832:
771:
764:
750:
707:
595:
518:
348:
262:
202:
82:
972:
778:
714:
404:
384:
321:
286:
207:
197:
182:
167:
121:
98:
30:
2861:
2568:
2561:
1045:
that permutes these three representations. The automorphism group acts on the center
697:
619:
453:
326:
192:
1129:
1030:
885:
552:
251:
240:
187:
162:
157:
116:
87:
50:
2757:
2782:
2235:
820:
19:
2557:
2550:
2150:
898:
881:
719:
447:
2718:
1502:
can be described as the product of bimultiplications of unit octonions, say
1125:
540:
58:
1154:
1015:
1011:
962:
77:
419:
333:
1935:
Multiplicative inverses of octonions are two-sided, which means that
1683:{\displaystyle \alpha =L_{\overline {u_{1}}}...L_{\overline {u_{n}}}}
1137:
1019:
1759:{\displaystyle \beta =R_{\overline {u_{1}}}...R_{\overline {u_{n}}}}
1140:. However the relationship is more complicated, partly due to the
1133:
18:
1095:. The triality symmetry acts again on the further quotient SO(8)/
1868:. As the set of orthogonal isotopies produce a 2-to-1 cover of
1128:, analogously to how elements of SO(2) can be described with
1084:, breaking this symmetry and obtaining SO(8), the remaining
2840:, Cambridge Studies in Advanced Mathematics, vol. 50,
1014:. This gives rise to peculiar feature of Spin(8) known as
2037:
can be permuted cyclically to give two further isotopies
1010:
under the Dynkin classification), possesses a three-fold
1607:{\displaystyle \alpha ,\beta \in \operatorname {SO(8)} }
2586:
2812:
The algebraic theory of spinors and
Clifford algebras
2580:
2502:
2451:
2400:
2349:
2298:
2247:
2211:
2191:
2159:
2123:
2081:
2043:
2005:
1973:
1941:
1906:
1874:
1830:
1810:
1772:
1696:
1620:
1574:
1508:
1488:
1448:
1414:
1357:
1319:
1287:
1249:
1211:
1171:
1077:≅GL(2,2)). When one quotients Spin(8) by one central
853:
490:
465:
428:
2564:
has 4! × 8 = 192 elements.
2149:. This "triality" automorphism is exceptional among
1145:unambiguously defined due to octonions obeying the
2756:John H. Conway; Derek A. Smith (23 January 2003).
2701:
2538:
2487:
2436:
2385:
2334:
2283:
2223:
2197:
2177:
2141:
2105:
2067:
2029:
1991:
1959:
1924:
1892:
1860:
1816:
1796:
1758:
1682:
1606:
1560:
1494:
1474:
1442:, the isotopy is called an orthogonal isotopy. If
1434:
1397:{\displaystyle x^{\alpha }y^{\beta }z^{\gamma }=1}
1396:
1343:
1305:
1273:
1235:
1197:
865:
498:
473:
436:
2814:, Collected works, vol. 2, Springer-Verlag,
2751:
2749:
1475:{\displaystyle \gamma \in \operatorname {SO(8)} }
1059:(which also has automorphism group isomorphic to
16:Rotation group in 8-dimensional Euclidean space
1561:{\displaystyle \gamma =B_{u_{1}}...B_{u_{n}}}
1243:, it can be shown that this is equivalent to
1124:Elements of SO(8) can be described with unit
923:, the diagonal matrices {±I} (as for all SO(2
800:
8:
1018:. Related to this is the fact that the two
2838:Clifford algebras and the classical groups
1861:{\displaystyle (-\alpha ,-\beta ,\gamma )}
1070:over the finite field with two elements,
807:
793:
245:
71:
36:
2581:
2579:
2501:
2450:
2399:
2348:
2297:
2246:
2231:are only uniquely determined up to sign.
2210:
2190:
2158:
2122:
2080:
2042:
2004:
1972:
1940:
1905:
1873:
1829:
1809:
1771:
1744:
1738:
1713:
1707:
1695:
1668:
1662:
1637:
1631:
1619:
1587:
1573:
1550:
1545:
1524:
1519:
1507:
1487:
1455:
1447:
1415:
1413:
1408:. If the three maps of an isotopy are in
1382:
1372:
1362:
1356:
1318:
1286:
1248:
1210:
1191:
1190:
1170:
852:
492:
491:
489:
467:
466:
464:
430:
429:
427:
2142:{\displaystyle \operatorname {Spin} (8)}
2106:{\displaystyle (\gamma ,\alpha ,\beta )}
2068:{\displaystyle (\beta ,\gamma ,\alpha )}
2030:{\displaystyle (\alpha ,\beta ,\gamma )}
1925:{\displaystyle \operatorname {Spin} (8)}
1797:{\displaystyle (\alpha ,\beta ,\gamma )}
1344:{\displaystyle (\alpha ,\beta ,\gamma )}
2745:
2153:. There is no triality automorphism of
835:. It could be either a real or complex
363:
129:
39:
2178:{\displaystyle \operatorname {SO} (8)}
1893:{\displaystyle \operatorname {SO} (8)}
1435:{\displaystyle \operatorname {SO(8)} }
1037:of Spin(8) which is isomorphic to the
847:Like all special orthogonal groups of
365:Classification of finite simple groups
1351:that preserve this identity, so that
1198:{\displaystyle x,y,z\in \mathbb {O} }
931:≥ 4), while the center of Spin(8) is
7:
1313:without ambiguity. A triple of maps
1066:which may also be considered as the
2789:, Chicago Lectures in Mathematics,
2787:Lectures on exceptional Lie groups
1999:. This means that a given isotopy
1766:. A simple calculation shows that
1591:
1588:
1459:
1456:
1419:
1416:
14:
2825:(originally published in 1954 by
2539:{\displaystyle (0,0,\pm 1,\pm 1)}
2488:{\displaystyle (0,\pm 1,0,\pm 1)}
2437:{\displaystyle (0,\pm 1,\pm 1,0)}
2386:{\displaystyle (\pm 1,0,0,\pm 1)}
2335:{\displaystyle (\pm 1,0,\pm 1,0)}
2284:{\displaystyle (\pm 1,\pm 1,0,0)}
997:
992:
987:
982:
977:
57:
2533:
2503:
2482:
2452:
2431:
2401:
2380:
2350:
2329:
2299:
2278:
2248:
2224:{\displaystyle \alpha ,\beta }
2172:
2166:
2136:
2130:
2100:
2082:
2062:
2044:
2024:
2006:
1919:
1913:
1887:
1881:
1855:
1831:
1791:
1773:
1600:
1594:
1468:
1462:
1428:
1422:
1338:
1320:
1262:
1253:
1221:
1212:
726:Infinite dimensional Lie group
1:
2759:On Quaternions and Octonions
1750:
1719:
1674:
1643:
839:of rank 4 and dimension 28.
831:acting on eight-dimensional
499:{\displaystyle \mathbb {Z} }
474:{\displaystyle \mathbb {Z} }
437:{\displaystyle \mathbb {Z} }
2791:University of Chicago Press
2113:. This produces an order 3
1482:, then following the above
1109:: Aut(Spin(8)) ≅ PSO (8) ⋊
224:List of group theory topics
2884:
2842:Cambridge University Press
967:SO(8) is unique among the
960:
2827:Columbia University Press
2762:. Taylor & Francis.
1086:outer automorphism group
1035:outer automorphism group
1033:of Spin(8) lives in the
829:special orthogonal group
342:Elementary abelian group
219:Glossary of group theory
2205:the corresponding maps
2198:{\displaystyle \gamma }
1900:, they must in fact be
1817:{\displaystyle \gamma }
1495:{\displaystyle \gamma }
1274:{\displaystyle x(yz)=1}
1236:{\displaystyle (xy)z=1}
2703:
2540:
2489:
2438:
2387:
2336:
2285:
2225:
2199:
2179:
2143:
2107:
2069:
2031:
1993:
1961:
1926:
1894:
1862:
1818:
1798:
1760:
1684:
1608:
1562:
1496:
1476:
1436:
1398:
1345:
1307:
1275:
1237:
1199:
1161:Octonions and triality
1136:can be described with
867:
866:{\displaystyle n>2}
758:Linear algebraic group
500:
475:
438:
34:
2704:
2541:
2490:
2439:
2388:
2337:
2286:
2226:
2200:
2180:
2144:
2108:
2070:
2032:
1994:
1992:{\displaystyle yzx=1}
1962:
1960:{\displaystyle xyz=1}
1927:
1895:
1863:
1819:
1799:
1761:
1685:
1609:
1563:
1497:
1477:
1437:
1399:
1346:
1308:
1306:{\displaystyle xyz=1}
1276:
1238:
1200:
868:
501:
476:
439:
22:
2578:
2500:
2449:
2398:
2347:
2296:
2245:
2209:
2189:
2157:
2121:
2079:
2041:
2003:
1971:
1939:
1904:
1872:
1828:
1808:
1770:
1694:
1618:
1572:
1506:
1486:
1446:
1412:
1355:
1317:
1285:
1247:
1209:
1169:
1130:unit complex numbers
1068:general linear group
851:
488:
463:
426:
132:Group homomorphisms
42:Algebraic structure
2699:
2693:
2536:
2485:
2434:
2383:
2332:
2281:
2221:
2195:
2175:
2139:
2115:outer automorphism
2103:
2065:
2027:
1989:
1957:
1922:
1890:
1858:
1814:
1794:
1756:
1680:
1604:
1558:
1492:
1472:
1432:
1394:
1341:
1303:
1271:
1233:
1195:
1147:Moufang identities
1107:semidirect product
945:(as for all Spin(4
863:
608:Special orthogonal
496:
471:
434:
315:Lagrange's theorem
35:
2808:Chevalley, Claude
2769:978-1-56881-134-5
2185:, as for a given
1967:is equivalent to
1753:
1722:
1677:
1646:
1142:non-associativity
1025:, as well as the
969:simple Lie groups
879:fundamental group
817:
816:
392:
391:
274:Alternating group
231:
230:
2875:
2854:
2834:Porteous, Ian R.
2824:
2803:
2774:
2773:
2753:
2724:Clifford algebra
2708:
2706:
2705:
2700:
2698:
2697:
2545:
2543:
2542:
2537:
2494:
2492:
2491:
2486:
2443:
2441:
2440:
2435:
2392:
2390:
2389:
2384:
2341:
2339:
2338:
2333:
2290:
2288:
2287:
2282:
2230:
2228:
2227:
2222:
2204:
2202:
2201:
2196:
2184:
2182:
2181:
2176:
2148:
2146:
2145:
2140:
2112:
2110:
2109:
2104:
2074:
2072:
2071:
2066:
2036:
2034:
2033:
2028:
1998:
1996:
1995:
1990:
1966:
1964:
1963:
1958:
1931:
1929:
1928:
1923:
1899:
1897:
1896:
1891:
1867:
1865:
1864:
1859:
1823:
1821:
1820:
1815:
1803:
1801:
1800:
1795:
1765:
1763:
1762:
1757:
1755:
1754:
1749:
1748:
1739:
1724:
1723:
1718:
1717:
1708:
1689:
1687:
1686:
1681:
1679:
1678:
1673:
1672:
1663:
1648:
1647:
1642:
1641:
1632:
1613:
1611:
1610:
1605:
1603:
1567:
1565:
1564:
1559:
1557:
1556:
1555:
1554:
1531:
1530:
1529:
1528:
1501:
1499:
1498:
1493:
1481:
1479:
1478:
1473:
1471:
1441:
1439:
1438:
1433:
1431:
1403:
1401:
1400:
1395:
1387:
1386:
1377:
1376:
1367:
1366:
1350:
1348:
1347:
1342:
1312:
1310:
1309:
1304:
1280:
1278:
1277:
1272:
1242:
1240:
1239:
1234:
1204:
1202:
1201:
1196:
1194:
1138:unit quaternions
1132:and elements of
1002:
1001:
1000:
996:
995:
991:
990:
986:
985:
981:
980:
897:of SO(8) is the
875:simply connected
872:
870:
869:
864:
837:simple Lie group
809:
802:
795:
751:Algebraic groups
524:Hyperbolic group
514:Arithmetic group
505:
503:
502:
497:
495:
480:
478:
477:
472:
470:
443:
441:
440:
435:
433:
356:Schur multiplier
310:Cauchy's theorem
298:Quaternion group
246:
72:
61:
48:
37:
2883:
2882:
2878:
2877:
2876:
2874:
2873:
2872:
2858:
2857:
2852:
2832:
2822:
2806:
2801:
2781:
2778:
2777:
2770:
2755:
2754:
2747:
2742:
2734:
2715:
2692:
2691:
2686:
2681:
2676:
2667:
2666:
2661:
2656:
2651:
2642:
2641:
2636:
2631:
2626:
2617:
2616:
2608:
2600:
2592:
2582:
2576:
2575:
2572:
2554:
2498:
2497:
2447:
2446:
2396:
2395:
2345:
2344:
2294:
2293:
2243:
2242:
2239:
2207:
2206:
2187:
2186:
2155:
2154:
2119:
2118:
2077:
2076:
2039:
2038:
2001:
2000:
1969:
1968:
1937:
1936:
1902:
1901:
1870:
1869:
1826:
1825:
1806:
1805:
1768:
1767:
1740:
1734:
1709:
1703:
1692:
1691:
1664:
1658:
1633:
1627:
1616:
1615:
1570:
1569:
1546:
1541:
1520:
1515:
1504:
1503:
1484:
1483:
1444:
1443:
1410:
1409:
1378:
1368:
1358:
1353:
1352:
1315:
1314:
1283:
1282:
1281:, meaning that
1245:
1244:
1207:
1206:
1167:
1166:
1163:
1122:
1115:
1101:
1094:
1083:
1076:
1065:
1058:
1051:
1044:
1039:symmetric group
1023:representations
1008:
998:
993:
988:
983:
978:
976:
965:
959:
944:
937:
922:
910:
895:universal cover
891:
873:, SO(8) is not
849:
848:
845:
833:Euclidean space
813:
784:
783:
772:Abelian variety
765:Reductive group
753:
743:
742:
741:
740:
691:
683:
675:
667:
659:
632:Special unitary
543:
529:
528:
510:
509:
486:
485:
461:
460:
424:
423:
415:
414:
405:Discrete groups
394:
393:
349:Frobenius group
294:
281:
270:
263:Symmetric group
259:
243:
233:
232:
83:Normal subgroup
69:
49:
40:
27:
17:
12:
11:
5:
2881:
2879:
2871:
2870:
2860:
2859:
2856:
2855:
2850:
2830:
2820:
2804:
2799:
2776:
2775:
2768:
2744:
2743:
2741:
2738:
2737:
2736:
2732:
2726:
2721:
2714:
2711:
2710:
2709:
2696:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2672:
2669:
2668:
2665:
2662:
2660:
2657:
2655:
2652:
2650:
2647:
2644:
2643:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2619:
2618:
2615:
2612:
2609:
2607:
2604:
2601:
2599:
2596:
2593:
2591:
2588:
2587:
2585:
2571:
2566:
2553:
2548:
2547:
2546:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2495:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2444:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2393:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2342:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2291:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2238:
2233:
2220:
2217:
2214:
2194:
2174:
2171:
2168:
2165:
2162:
2138:
2135:
2132:
2129:
2126:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
1988:
1985:
1982:
1979:
1976:
1956:
1953:
1950:
1947:
1944:
1921:
1918:
1915:
1912:
1909:
1889:
1886:
1883:
1880:
1877:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1813:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1752:
1747:
1743:
1737:
1733:
1730:
1727:
1721:
1716:
1712:
1706:
1702:
1699:
1676:
1671:
1667:
1661:
1657:
1654:
1651:
1645:
1640:
1636:
1630:
1626:
1623:
1602:
1599:
1596:
1593:
1590:
1586:
1583:
1580:
1577:
1553:
1549:
1544:
1540:
1537:
1534:
1527:
1523:
1518:
1514:
1511:
1491:
1470:
1467:
1464:
1461:
1458:
1454:
1451:
1430:
1427:
1424:
1421:
1418:
1393:
1390:
1385:
1381:
1375:
1371:
1365:
1361:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1302:
1299:
1296:
1293:
1290:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1193:
1189:
1186:
1183:
1180:
1177:
1174:
1162:
1159:
1121:
1120:Unit octonions
1118:
1113:
1099:
1092:
1081:
1074:
1063:
1056:
1049:
1042:
1006:
973:Dynkin diagram
961:Main article:
958:
955:
942:
935:
920:
909:
906:
889:
862:
859:
856:
844:
841:
815:
814:
812:
811:
804:
797:
789:
786:
785:
782:
781:
779:Elliptic curve
775:
774:
768:
767:
761:
760:
754:
749:
748:
745:
744:
739:
738:
735:
732:
728:
724:
723:
722:
717:
715:Diffeomorphism
711:
710:
705:
700:
694:
693:
689:
685:
681:
677:
673:
669:
665:
661:
657:
652:
651:
640:
639:
628:
627:
616:
615:
604:
603:
592:
591:
580:
579:
572:Special linear
568:
567:
560:General linear
556:
555:
550:
544:
535:
534:
531:
530:
527:
526:
521:
516:
508:
507:
494:
482:
469:
456:
454:Modular groups
452:
451:
450:
445:
432:
416:
413:
412:
407:
401:
400:
399:
396:
395:
390:
389:
388:
387:
382:
377:
374:
368:
367:
361:
360:
359:
358:
352:
351:
345:
344:
339:
330:
329:
327:Hall's theorem
324:
322:Sylow theorems
318:
317:
312:
304:
303:
302:
301:
295:
290:
287:Dihedral group
283:
282:
277:
271:
266:
260:
255:
244:
239:
238:
235:
234:
229:
228:
227:
226:
221:
213:
212:
211:
210:
205:
200:
195:
190:
185:
180:
178:multiplicative
175:
170:
165:
160:
152:
151:
150:
149:
144:
136:
135:
127:
126:
125:
124:
122:Wreath product
119:
114:
109:
107:direct product
101:
99:Quotient group
93:
92:
91:
90:
85:
80:
70:
67:
66:
63:
62:
54:
53:
31:Dynkin diagram
25:
15:
13:
10:
9:
6:
4:
3:
2:
2880:
2869:
2866:
2865:
2863:
2853:
2851:0-521-55177-3
2847:
2843:
2839:
2835:
2831:
2828:
2823:
2821:3-540-57063-2
2817:
2813:
2809:
2805:
2802:
2800:0-226-00526-7
2796:
2792:
2788:
2784:
2780:
2779:
2771:
2765:
2761:
2760:
2752:
2750:
2746:
2739:
2735:
2731:
2727:
2725:
2722:
2720:
2717:
2716:
2712:
2694:
2688:
2683:
2678:
2673:
2670:
2663:
2658:
2653:
2648:
2645:
2638:
2633:
2628:
2623:
2620:
2613:
2610:
2605:
2602:
2597:
2594:
2589:
2583:
2574:
2573:
2570:
2569:Cartan matrix
2567:
2565:
2563:
2562:Coxeter group
2559:
2552:
2549:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2496:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2445:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2394:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2343:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2292:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2241:
2240:
2237:
2234:
2232:
2218:
2215:
2212:
2192:
2169:
2163:
2160:
2152:
2133:
2127:
2124:
2116:
2097:
2094:
2091:
2088:
2085:
2059:
2056:
2053:
2050:
2047:
2021:
2018:
2015:
2012:
2009:
1986:
1983:
1980:
1977:
1974:
1954:
1951:
1948:
1945:
1942:
1933:
1916:
1910:
1907:
1884:
1878:
1875:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1811:
1788:
1785:
1782:
1779:
1776:
1745:
1741:
1735:
1731:
1728:
1725:
1714:
1710:
1704:
1700:
1697:
1669:
1665:
1659:
1655:
1652:
1649:
1638:
1634:
1628:
1624:
1621:
1597:
1584:
1581:
1578:
1575:
1551:
1547:
1542:
1538:
1535:
1532:
1525:
1521:
1516:
1512:
1509:
1489:
1465:
1452:
1449:
1425:
1407:
1404:is called an
1391:
1388:
1383:
1379:
1373:
1369:
1363:
1359:
1335:
1332:
1329:
1326:
1323:
1300:
1297:
1294:
1291:
1288:
1268:
1265:
1259:
1256:
1250:
1230:
1227:
1224:
1218:
1215:
1187:
1184:
1181:
1178:
1175:
1172:
1160:
1158:
1156:
1150:
1148:
1143:
1139:
1135:
1131:
1127:
1119:
1117:
1112:
1108:
1103:
1098:
1091:
1087:
1080:
1073:
1069:
1062:
1055:
1048:
1040:
1036:
1032:
1028:
1024:
1021:
1017:
1013:
1009:
974:
970:
964:
956:
954:
952:
948:
941:
934:
930:
926:
919:
915:
907:
905:
903:
900:
896:
892:
888:
883:
880:
876:
860:
857:
854:
842:
840:
838:
834:
830:
826:
822:
810:
805:
803:
798:
796:
791:
790:
788:
787:
780:
777:
776:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
747:
746:
736:
733:
730:
729:
727:
721:
718:
716:
713:
712:
709:
706:
704:
701:
699:
696:
695:
692:
686:
684:
678:
676:
670:
668:
662:
660:
654:
653:
649:
645:
642:
641:
637:
633:
630:
629:
625:
621:
618:
617:
613:
609:
606:
605:
601:
597:
594:
593:
589:
585:
582:
581:
577:
573:
570:
569:
565:
561:
558:
557:
554:
551:
549:
546:
545:
542:
538:
533:
532:
525:
522:
520:
517:
515:
512:
511:
483:
458:
457:
455:
449:
446:
421:
418:
417:
411:
408:
406:
403:
402:
398:
397:
386:
383:
381:
378:
375:
372:
371:
370:
369:
366:
362:
357:
354:
353:
350:
347:
346:
343:
340:
338:
336:
332:
331:
328:
325:
323:
320:
319:
316:
313:
311:
308:
307:
306:
305:
299:
296:
293:
288:
285:
284:
280:
275:
272:
269:
264:
261:
258:
253:
250:
249:
248:
247:
242:
241:Finite groups
237:
236:
225:
222:
220:
217:
216:
215:
214:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
179:
176:
174:
171:
169:
166:
164:
161:
159:
156:
155:
154:
153:
148:
145:
143:
140:
139:
138:
137:
134:
133:
128:
123:
120:
118:
115:
113:
110:
108:
105:
102:
100:
97:
96:
95:
94:
89:
86:
84:
81:
79:
76:
75:
74:
73:
68:Basic notions
65:
64:
60:
56:
55:
52:
47:
43:
38:
32:
28:
21:
2837:
2811:
2786:
2758:
2729:
2555:
1934:
1164:
1151:
1123:
1110:
1104:
1096:
1089:
1078:
1071:
1060:
1053:
1046:
1031:automorphism
971:in that its
966:
950:
946:
939:
932:
928:
924:
917:
916:of SO(8) is
911:
901:
886:
846:
824:
818:
647:
635:
623:
611:
599:
587:
575:
563:
334:
291:
278:
267:
256:
252:Cyclic group
130:
117:Free product
88:Group action
51:Group theory
46:Group theory
45:
2783:Adams, J.F.
2236:Root system
2151:spin groups
1027:fundamental
877:, having a
821:mathematics
537:Topological
376:alternating
2868:Lie groups
2740:References
2551:Weyl group
899:spin group
882:isomorphic
644:Symplectic
584:Orthogonal
541:Lie groups
448:Free group
173:continuous
112:Direct sum
2719:Octonions
2671:−
2646:−
2621:−
2611:−
2603:−
2595:−
2528:±
2519:±
2477:±
2462:±
2420:±
2411:±
2375:±
2354:±
2318:±
2303:±
2261:±
2252:±
2219:β
2213:α
2193:γ
2164:
2128:
2098:β
2092:α
2086:γ
2060:α
2054:γ
2048:β
2022:γ
2016:β
2010:α
1911:
1879:
1853:γ
1847:β
1844:−
1838:α
1835:−
1812:γ
1789:γ
1783:β
1777:α
1751:¯
1720:¯
1698:β
1675:¯
1644:¯
1622:α
1585:∈
1582:β
1576:α
1510:γ
1490:γ
1453:∈
1450:γ
1384:γ
1374:β
1364:α
1336:γ
1330:β
1324:α
1188:∈
1126:octonions
708:Conformal
596:Euclidean
203:nilpotent
2862:Category
2836:(1995),
2810:(1997),
2785:(1996),
2713:See also
1155:triality
1088:is only
1016:triality
1012:symmetry
963:Triality
957:Triality
927:) with 2
703:Poincaré
548:Solenoid
420:Integers
410:Lattices
385:sporadic
380:Lie type
208:solvable
198:dihedral
183:additive
168:infinite
78:Subgroup
33:of SO(8)
1406:isotopy
902:Spin(8)
843:Spin(8)
827:is the
698:Lorentz
620:Unitary
519:Lattice
459:PSL(2,
193:abelian
104:(Semi-)
2848:
2818:
2797:
2766:
1568:. Let
1020:spinor
953:≥ 4).
938:×
914:center
908:Center
893:. The
553:Circle
484:SL(2,
373:cyclic
337:-group
188:cyclic
163:finite
158:simple
142:kernel
1134:SO(4)
825:SO(8)
737:Sp(∞)
734:SU(∞)
147:image
2846:ISBN
2816:ISBN
2795:ISBN
2764:ISBN
2558:Weyl
2556:Its
2125:Spin
2075:and
1908:Spin
1205:and
949:), 4
912:The
858:>
731:O(∞)
720:Loop
539:and
2117:of
1824:is
1165:If
1149:).
884:to
819:In
646:Sp(
634:SU(
610:SO(
574:SL(
562:GL(
2864::
2844:,
2793:,
2748:^
2161:SO
1932:.
1876:SO
1690:,
1116:.
1102:.
1052:x
975:,
904:.
823:,
622:U(
598:E(
586:O(
44:→
29:,
2829:)
2772:.
2733:2
2730:G
2695:)
2689:2
2684:0
2679:0
2674:1
2664:0
2659:2
2654:0
2649:1
2639:0
2634:0
2629:2
2624:1
2614:1
2606:1
2598:1
2590:2
2584:(
2560:/
2534:)
2531:1
2525:,
2522:1
2516:,
2513:0
2510:,
2507:0
2504:(
2483:)
2480:1
2474:,
2471:0
2468:,
2465:1
2459:,
2456:0
2453:(
2432:)
2429:0
2426:,
2423:1
2417:,
2414:1
2408:,
2405:0
2402:(
2381:)
2378:1
2372:,
2369:0
2366:,
2363:0
2360:,
2357:1
2351:(
2330:)
2327:0
2324:,
2321:1
2315:,
2312:0
2309:,
2306:1
2300:(
2279:)
2276:0
2273:,
2270:0
2267:,
2264:1
2258:,
2255:1
2249:(
2216:,
2173:)
2170:8
2167:(
2137:)
2134:8
2131:(
2101:)
2095:,
2089:,
2083:(
2063:)
2057:,
2051:,
2045:(
2025:)
2019:,
2013:,
2007:(
1987:1
1984:=
1981:x
1978:z
1975:y
1955:1
1952:=
1949:z
1946:y
1943:x
1920:)
1917:8
1914:(
1888:)
1885:8
1882:(
1856:)
1850:,
1841:,
1832:(
1792:)
1786:,
1780:,
1774:(
1746:n
1742:u
1736:R
1732:.
1729:.
1726:.
1715:1
1711:u
1705:R
1701:=
1670:n
1666:u
1660:L
1656:.
1653:.
1650:.
1639:1
1635:u
1629:L
1625:=
1601:)
1598:8
1595:(
1592:O
1589:S
1579:,
1552:n
1548:u
1543:B
1539:.
1536:.
1533:.
1526:1
1522:u
1517:B
1513:=
1469:)
1466:8
1463:(
1460:O
1457:S
1429:)
1426:8
1423:(
1420:O
1417:S
1392:1
1389:=
1380:z
1370:y
1360:x
1339:)
1333:,
1327:,
1321:(
1301:1
1298:=
1295:z
1292:y
1289:x
1269:1
1266:=
1263:)
1260:z
1257:y
1254:(
1251:x
1231:1
1228:=
1225:z
1222:)
1219:y
1216:x
1213:(
1192:O
1185:z
1182:,
1179:y
1176:,
1173:x
1114:3
1111:S
1100:2
1097:Z
1093:2
1090:Z
1082:2
1079:Z
1075:3
1072:S
1064:3
1061:S
1057:2
1054:Z
1050:2
1047:Z
1043:3
1041:S
1007:4
1005:D
1003:(
951:n
947:n
943:2
940:Z
936:2
933:Z
929:n
925:n
921:2
918:Z
890:2
887:Z
861:2
855:n
808:e
801:t
794:v
690:8
688:E
682:7
680:E
674:6
672:E
666:4
664:F
658:2
656:G
650:)
648:n
638:)
636:n
626:)
624:n
614:)
612:n
602:)
600:n
590:)
588:n
578:)
576:n
566:)
564:n
506:)
493:Z
481:)
468:Z
444:)
431:Z
422:(
335:p
300:Q
292:n
289:D
279:n
276:A
268:n
265:S
257:n
254:Z
26:4
24:D
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