So I stumbled across the isomorphism theorems for Lie Algebras which are as follows:
Let $L_1$ and $L_2$ be Lie algebras, $\phi$ : $L_1$ $\rightarrow$ $L_2$ a homomorphism of Lie algebras and $I$,$J$ ideals of a Lie algebra $L$ $\Rightarrow$
1.) $L_1/$Ker($\phi$) $\cong$ im($\phi$)
2.) $(I+J)/J$ $\cong$ $I/(I \cap J)$\
3.) if $I \subseteq J$ $\Rightarrow$ $J/I$ is an ideal of $L/I$ and $(L/I)/(J/I)$ $\cong$ $L/J$
That these theorems hold for vector spaces and their subspaces is already known.I read somewhere that because the canonical projection
$\pi$: $L \rightarrow L/I$, $x \mapsto x+I$ is a homomorphism of Lie algebras that these theorems are also true for Lie algebras.
I don't see how we can conclude that from $\pi$ being a homomorphism.
Hints: The crucial thing for $\pi$ being Lie algebra homomorphism, is that the induced Lie bracket $[x+I, \, y+I] :=[x,y] +I$ on $L/I$ is well defined.
For the first isomorphism theorem, we obtain a vector space isomorphism $\phi':L_1/\ker\phi\to {\rm im}\, \phi$, and since $\ker\phi$ is an ideal, $L_1/\ker\phi$ is a Lie algebra, and by construction $\phi'$ carries the induced Lie bracket of $L_1/\ker\phi$ to the one of $L_2$, so $\phi'$ is also a Lie algebra isomorphism.
The other statements follow from the first isomorphism theorem in the usual way.