Toral subalgebra

Question

In the proof that a Toral subalgebra $T$ of a semisimple Lie algebra $L$ is abelian, one argues that if $x\in T$ is ad-simple ($ad_x:L\to L$ is diagonalizable), the restriction $ad_x|_T:T\to T$ is also diagonalizable. Why this is true? I can follow the rest of he argument.

Answer 1

This is an instance of a more general fact from linear algebra:

Let $T : V \to V$ be a linear map, with $U \subseteq V$ a $T$-invariant subspace, so that $T(U)\subseteq U$. Then $T_{\big \vert U}$ is diagonalisable whenever $T$ is.

You can find multiple proofs of this fact here:

Diagonalizable transformation restricted to an invariant subspace is diagonalizable

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Note that in our case, that $T$ is an invariant subspace under $\operatorname{ad}_x$ for each $x \in T$ is precisely the condition that $T$ is a subalgebra of $L$.