How to verify the Jacobi identity for the semidirect product Lie algebra

Question

I've been trying to check the claim that the vector space direct sum $L \oplus D$ is a Lie algebra, and I'm having a lot of trouble with verifying the Jacobi identity. It's defined where $L$ is a Lie algebra and $D$ is a subalgebra of Der(L) with bracket $[x_1 + d_1, x_2 + d_2] = [x_1,x_2] + d_1(x_2) - d_2(x_1) + [d_1,d_2]$.

I started with $[x,[y,z]] + [y,[z,x]] + [z,[x,y]]$ for $x = x_1 + d_1, y = x_2 + d_2, z = x_3 + d_3$, with the aim of showing that this simplifies to 0 by bilinearity and definition of the bracket. I simplified all the inner brackets first, but then get confused when trying to break down further. I end up getting what seem like strange compositions of elements of L and derivations.

Answer 1

Linearity of the Lie product in each component, and the cyclic format of the Jacobi identify, allows you to deal with four cases.

1. $x, y, z \in L$, that's Jacobi in $L$.

2. $x, y, z \in D$, that's Jacobi in $D$.

3. $x, y\in L$, and $z \in D$, that's the fact that the elements of $D$ are derivations of $L$.

4. $x\in L$, and $y, z \in D$, that's the fact that $D$ is a Lie subalgebra of $\operatorname{Der}(L) $.