$i_1,i_2$ are ideals of Lie algebra $\mathfrak{g}$, is $[i_1,i_2]$ also an ideal?

Question

If $i_1$ and $i_2$ are ideals of $\mathfrak{g}$, I want to show that $[i_1,i_2]$ is also an ideal of $\mathfrak{g}$.

This is how I proceed:

$[i_1,\mathfrak{g}]\subseteq i_1,[i_2,\mathfrak{g}]\subseteq i_2$

We want $[[i_1,i_2],\mathfrak{g}]\subseteq [i_1,i_2]$:

$$[[i_1,i_2],\mathfrak{g}]= -[\mathfrak{g},[i_1,i_2]]=[i_1,[i_2,\mathfrak{g}]]+ [i_2,[\mathfrak{g},i_1]]=[i_1,[i_2,\mathfrak{g}]]+ [i_2,-[i_1,\mathfrak{g}]]$$ $$\subseteq [i_1,i_2]+ [i_2,-i_1]=[i_1,i_2]+ [i_1,i_2]=2[i_1,i_2]=[i_1,i_2]$$

Is this acceptable?

Answer 1

Use Jacobi. You have to show that for every $i_1\in I_1, i_2\in I_2$ and $x\in {\cal G}$, $[x,[i_1,i_2]]\in [I_1,I_2]$. Jacobi implies that $[x,[i_1,i_2]]+[i_2,[x,i_1]]+[i_1,[i_2,x]]=0$, thus $[x,[i_1,i_2]] =-[i_2,[x,i_1]]-[i_1,[i_2,x]]$

$[x,i_1]\in I_1$ since $I_1$ is an ideal, thus $[i_2,[x,i_1]]\in [I_1,I_2]$

$[i_2,x]\in I_2$ since $I_2$ is an ideal, thus $[i_1,[i_2,x]]\in [I_1,I_2]$