Verifying the Jacobi identity for the semidirect product of Lie algebras

Given Lie algebras $S$ and $I$ and a Lie homomorphism $\theta \colon S\to \operatorname{Der} I$, we have the semidirect product to be the vector space $S\oplus I$ with operation $$ (s_{1},x_{1})(s_{2}x_{2}) := ([s_{1},s_{2}],[x_{1},x_{2}]+\theta(s_{1})x_{2}-\theta(s_{2})x_{1}). $$ Show that this is a Lie algebra.

So I can easily verify the skew-symmetric but I can't seem to work out a nice way of proving the Jacobi identity. Am I missing a simple trick or must you perform the tedious calculation to show this? Thanks.

Solution 1:

The calculation is no longer tedious if you split it up into four cases. Since the Jacobi identity is trilinear we only need to check it for one of the following cases: $(s_1,0),(s_2,0),(s_3,0)$, or $(s_1,0),(s_2,0),(0,x_3)$, or $(s_1,0),(0,x_2),(0,x_3)$ or $(0,x_1),(0,x_2),(0,x_3)$. The cases themselves are immediate, because they follow from the facts that either $S$ is a Lie algebra, or that $I$ is a Lie algebra, or that the $\theta(s_i)$ are derivations, or that $\theta$ is a Lie algebra homomorphism.