= Solution
No. Let $\mathfrak g=\mathfrak{sl}_2$, let $\mathfrak s=\mathbb Ce$ be the one-dimensional subalgebra generated by the standard raising operator, and let $L=\mathbb C^2$ be the irreducible defining representation. On restriction to $\mathfrak s$, the element $e$ acts by a nonzero <Nilpotent Jordan block>. A direct sum of irreducible representations of the one-dimensional abelian Lie algebra would make $e$ diagonalizable, so this restriction is not completely reducible. The <Complete reducibility of semisimple Lie algebra representations> applies when the restricting algebra is semisimple, which $\mathfrak s$ is not.
Back to article page
