Let $L$ be a Lie algebra and $A$ be its subalgebra. Let consider $h : A \to Der(L)$ defined by $h(a)(l)=[l,a]$ then how to show that $h$ is Lie algebra homomorphism?
According to the definition of Lie algebras homomorphisms $h$ must satisfies $ h([a_1,a_2])=[h(a_1),h(a_2)]$. Using Jacobi identity, we need to prove that $[[l_1,l_2]a]$ equals to $[[l_1,a],[l_2,a]]$ ? I have problem with calculation of Jacobi identity!
Edit: There are useful answers to this question: Checking a Lie homomorphism has been described carefully in Adjoint map is Lie homomorphism and Adjoint map is a Lie homomorphism.