= Solution
Let $N=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)$ and $\mathfrak g=\mathbb C(I+N)\subseteq\mathfrak{gl}_2(\mathbb C)$. This one-dimensional <abelian Lie algebra> is not a <semisimple Lie algebra>. For $X=I+N$, its <Additive Jordan decomposition> is $X_s=I$ and $X_n=N$. Neither belongs to $\mathfrak g$: comparing diagonal entries and then the upper-right entry rules out both scalar multiples. Therefore \b[nonsemisimple matrix <Lie subalgebras> need not be closed under <Additive Jordan decomposition>].
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