 Solution

No. Let $\mathfrak{g}={\mathfrak{sl}}_2$, let $\mathfrak{s}=\mathbb{C}e$ 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.