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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.

Nil
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  • Many Thanks. Is it true to follow this way to show Leibniz homomorphisms? – Nil Apr 26 '18 at 14:43
  • For right Leibniz algebra $L$, the right multiplication is a derivation, so if we define $h: L \to Der(L)$ then how to prove it is Leibniz homomorphism? – Nil Apr 26 '18 at 14:46
  • I found my old proof again, see here. For Leibniz you have to rewrite Jacobi either to "left Leibniz", or "right Leibniz". – Dietrich Burde Apr 26 '18 at 14:58
  • Your old proof is clear and useful. In the case of Leibniz algebras, if we write $h: A \to Der(L)$ where $A$ is a subalgebra of $L$, then still it works? Means is it Leibniz homomorphism? – Nil Apr 26 '18 at 15:03
  • Yes, the adjoint map is a Leibniz algebra representation. – Dietrich Burde Apr 26 '18 at 15:10

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