# Definition A *Lie algebra homomorphism* from a [[Lie algebra]] $\mathfrak{g}$ to a Lie algebra $\mathfrak{h}$ is a [[linear map]] $\phi: \mathfrak{g} \rightarrow \mathfrak{h}$ that preserves the Lie bracket in the following way $ [\phi(X), \phi(Y)] = \phi([X,Y]) \quad \forall X, Y \in \mathfrak{g} $ Note that the commutator on the left-hand side takes place in $\mathfrak{h}$ while on the right hand side it takes place in $\mathfrak{g}$.