Consider $g$ a Lie algebra.
Prove that if $ad(g)$ is semisimples then the $ad (g)$ representation is completely reducible.
Prove: if ad g is semi-simple, an i apply the Weyl's theorem directly to say that g is completely reducible $ad (g)$-modulus?
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1Hint: Look at the complexification of $\mathcal{g}$ (if necessary) and then look at the compact real form! – Brozovic Mar 31 '21 at 12:43
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Not "lie algebra" but Lie algebra, like Weyl's theorem (not weyl's theorem). – Dietrich Burde Apr 02 '21 at 08:30