Prove $[\mathfrak g,\mathfrak g]$ is an ideal.

Question

I have to show :

Given a Lie algebra $\mathfrak g$, then $[\mathfrak g,\mathfrak g]$ is an ideal.

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I was told to use Jacobi's identity, but I am not sure why.

It seems I just have to show that for $x,y,z \in \mathfrak g$, I have $[x,[y,z]]\in [\mathfrak g,\mathfrak g]$ which is the case since $x\in \mathfrak g$ and $[x,y]\in \mathfrak g$ since $\mathfrak g$ is a Lie algebra so closed under the bracket.

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What am I getting wrong here?

Answer 1

Your argument seems to be correct. Indeed, $[y,z]\in \mathfrak{g}$ so that $[x,[y,z]]\in [\mathfrak{g},\mathfrak{g}]$. $[\mathfrak{g},\mathfrak{g}]$ consists of elements of the form $$ \sum_{i=1}^na_i[y_i,z_i]$$ then $$ \bigg[x,\sum_{i=1}^na_i[y_i,z_i]\bigg]=\sum_{i=1}^na_i[x,[y_i,z_i]]\in [\mathfrak{g},\mathfrak{g}].$$ In short, I believe your argument is fine.