Is the image of an ideal under a surjective Lie algebra homomorphism an ideal?

Question

Say we have $\phi: \mathfrak{g} \to \mathfrak{h}$, where $\phi$ is a surjective Lie algebra homomorphism. Is $\phi(I)$ an ideal of $\mathfrak{h}$? I think this holds because for every element of $h \in \mathfrak{h}$ we may consider $\phi^{-1}(h) \subset \mathfrak{g}$, and in this way inherit the ideal structure, but I just wanted to make sure.

Answer 1

Since it's a linear map, the image is a subspace. To see that it is an ideal, if $a \in \mathfrak{h},b \in \phi(I)$. Since $\phi$ is surjective, find $\alpha \in \mathfrak{g}$ such that $\phi(\alpha)=a$ and $\beta \in I$ such that $\phi(\beta)=b$. We then get that

$[a,b] = [\phi(\alpha),\phi(\beta)] = \phi([\alpha, \beta]) \in \phi(I)$

since $[\alpha, \beta] \in I$ since $I$ is an ideal.

Also, this is true for any algebras, not just Lie algebras.