 Solution (source code)

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= Solution

Let $W\subseteq\mathfrak g$ be invariant under the <Adjoint representation of a Lie algebra>. Then $[\mathfrak g,W]\subseteq W$, so $W$ is an ideal. If $\mathfrak g$ is a <Simple Lie algebra>, its only ideals are $0$ and $\mathfrak g$. Hence the adjoint representation is irreducible. Compact type is compatible with the anti-Hermitian realization used below, although simplicity alone proves this <adjoint irreducibility of a simple Lie algebra>.

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