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Thank you all for contributing to this entry. Here are some suggestions: (1) The Dynkin index is defined for any representation of a simple Lie algebra (there is nothing called "compact Lie algebra"). (2) The notation used $ x_\lambda$ is not common in math but might be in some physics book. (3)
166:(3a) a representation, (3b) a simple subalgebra (up to linear equivalence) of a simple Lie algebra, (3c) and also for just a simple Lie algebra (as the common divisors of the Dynkin indices of its fundamental representations).
169:(4) In the definition using the Killing form, it is very important to normalize the trace so that a long root has length square equals to 2. (5) adding refs to the original paper of Dynkin.
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