I always thought of the Killing form as the natural way to introduce an invariant inner product (that was nontrivial) on a Lie algebra, hence the ideal tools for proving theorems in the semisimple case. Indeed, the trace form is a reasonable invariant bilinear form on $\mathfrak{gl}_n$, and the adjoint map is the first choice of a map from a Lie algebra into $\mathfrak{gl}_n$ that one would think of.
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I always thought of the Killing form as the natural way to introduce an invariant inner product (that was nontrivial) on a Lie algebra, hence the ideal tools for proving theorems in the semisimple case. |
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