Solvable Lie algebra and its Center

Question

If $\mathfrak{g}$ is a solvable Lie algebra with center $Z(\mathfrak{g})$, can we decompose it over its center? Writing $\mathfrak{g}=Z(\mathfrak{g})\oplus\mathfrak{h}$ where $\mathfrak{h}$ is a Lie subalgebra?

Answer 1

In general a solvable Lie algebra does not decompose as a Lie algebra direct sum with the center. Take the $3$-dimensional Heisenberg Lie algebra $L$ with basis $(x,y,z)$ and $[x,y]=z$, $Z(L)=\langle z\rangle$. If you consider this sum just as direct sum of vector spaces, then we always have $$\mathfrak{g}=Z(\mathfrak{g})\oplus \mathfrak{h}=Z(\mathfrak{g})\oplus\mathfrak{g}/Z(\mathfrak{g}).$$