= Positive trace index for a compact simple Lie algebra
{title2=$\operatorname{Tr}(d(X)d(Y))=\mu\kappa(X,Y),\quad\mu>0$}
For a nontrivial finite-dimensional <anti-Hermitian> <Lie algebra representation> of a <compact Lie algebra> that is simple, its <Trace form of a Lie algebra representation> is a positive multiple of the <Killing form>. Define $T$ by $H(X,Y)=\kappa(TX,Y)$. Invariance makes $T$ commute with the <Adjoint representation of a Lie algebra>. Symmetry makes $T$ self-adjoint for $-\kappa$, so its <eigenspaces> are ideals. Simplicity implies $T=\mu I$. The kernel of the representation is an ideal and hence vanishes; therefore $H(X,X)=-\operatorname{Tr}(d(X)^\dagger d(X))<0$ for nonzero $X$, proving $\mu>0$.
// Target: lie-theory.bigb
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