Splitting of Lie algebra extensions: why does this linear map exist?

Question

Let $\mathfrak{b}$ and $\mathfrak{g}$ be finite dimensional Lie algebras and let $(\tilde{\mathfrak{g}},j,\phi)$ be a Lie algebra extension of $\mathfrak{g}$ by $\mathfrak{b}$, so we have a short exact sequence of Lie algebras$$ 0\to\mathfrak{b}\xrightarrow{\;\;j\;\;}\tilde{\mathfrak{g}}\xrightarrow{\;\;\phi\;\;}\mathfrak{g}\to0 $$ as in page 15 of these notes by Erik van den Ban. Then, the author chooses a linear map $\xi:\mathfrak{g}\to\tilde{\mathfrak{g}}$ such that $\phi\circ\xi=Id_{\mathfrak{g}}$. If $\xi$ is a Lie algebra homomorphism, it means that the central extension splits but it is just assumed to be linear. Why does this map exist? I read on here that every short exact sequence of vector spaces splits, is there an intuitive reason why that is true?

Answer 1

Extensions of vector spaces split, but not extensions of Lie algebras in general. So "this homomorphism $\xi$" need not exist. Consider an example, i.e., a non-split extension of the $3$-dimensional Heisenberg Lie algebra:

Non-split extension of Lie algebras?

For vector spaces, or more generally for $R$-modules see here:

Why any short exact sequence of vector spaces may be seen as a direct sum?

A question about split short exact sequence of modules