I am trying to show that a complex Lie algebra $\mathfrak g$ is the direct sum of simple ideals iff it is semisimple.
In fact, I have already proved that it is sufficient. I am now trying to prove that $\mathfrak g$ being a direct sum of simple ideals implies that it is semisimple.
Currently, I would like to prove that if $J$ is an ideal of the direct sum of simple ideals, then $J$ is itself a direct sum of some subset of these simple ideals.
So considering $J$ to be a direct sum of $L_i$, I am able to show that if $J$ is not equal to the direct sum of all the $L_i$, then there exists $i$ such that $[L_i,J]=0$.
From this how can I conclude that $J$ is contained in the direct sum of the $L_j$s with $j\neq i$?