Image of a direct sums of Lie algebras

Question

Let $\mathfrak{g}$ be a Lie algebra with ideals $\mathfrak{h}$ and $\mathfrak{k}$ such that $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{k}$. If $\phi:\mathfrak{g}\to\phi(\mathfrak{g})$ is a Lie algebra homomorphism, is it true that $\phi(\mathfrak{g})=\phi(\mathfrak{h})+\phi(\mathfrak{k})$, but not a direct sum anymore, and that it is a direct sum of lie algebras if and only if $\phi$ is injective.

Answer 1

It is entirely possible for $\phi$ not to be injective and yet $\phi(\mathfrak h)=\phi(\mathfrak h)\oplus\phi(\mathfrak k)$: pick your favourite non-injective (and non-zero) homomorphisms $\psi:\mathfrak k\to\mathfrak k$ and $\rho:\mathfrak h\to \mathfrak h$ and call $$\phi:\mathfrak h\oplus \mathfrak k\to\mathfrak h\oplus \mathfrak k\\ \phi(v,w)=(\rho(v),\psi(w))$$

It is true in general that $\phi(\mathfrak g)=\phi(\mathfrak h)+\phi( \mathfrak k)$, because it is true a fortiori for vector subspaces and linear maps.