Lie-algebra homomorphism

Let $\mathfrak{g}$ and $\mathfrak{h}$ be finite-dimensional Lie algebras over the same field $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$. A Lie-algebra homomorphism

$\phi :\mathfrak{g}⟶\mathfrak{h}$

is an $\mathbb{F}$-linear map, in the sense of Linear map between vector spaces over the same field, such that

$\phi ([X,Y{]}_{\mathfrak{g}})=[\phi (X),\phi (Y){]}_{\mathfrak{h}}$

for every $X,Y\in \mathfrak{g}$. Both linearity and bracket preservation are requirements. For complex Lie algebras, $\phi$ must therefore be complex-linear, not merely real-linear.

The zero brackets allowed by Finite-dimensional Lie algebra are not required to be nondegenerate. The unique linear map from the zero Lie algebra to any $\mathfrak{h}$ is a homomorphism. One-dimensional Lie algebras have zero bracket, so a linear map between two such algebras preserves the bracket automatically, while a linear map from an abelian algebra to a nonabelian one still has to have commuting image.

The source and target are nonempty because they contain zero. The supplied map is data, and checking the two universal algebraic identities uses no choice. There is no metric, manifold boundary, interval, or endpoint. This is a definition with two simultaneous requirements, not a biconditional theorem.