Is there any good example about Lie algebra homomorphisms? My textbook gave an example of the trace, but I think to get a better comprehension, more examples are still needed.
Any example will be helpful ~

A good source of examples is the free Lie algebra $\mathcal L(n)$, on generators $X_1,\ldots, X_n$. This is defined as the vector space with basis given by all formal bracketing expressions of generators, such as $[X_1,X_2]$, $[X_3+2X_4,[X_5,X_7]]$, etc. One the takes the quotient by relators representing antisymmetry of the bracket, the Jaobi identity and multilinearity.
This gives a lot of homomorphisms. For example, $\mathcal L(n)$ maps homomorphically onto $\mathcal L(n-1)$ where the homomorphism is defined by setting a variable equal to $0$. Similarly $\mathcal L(n)$ maps to $\mathcal L(n+1)$ by inclusion, which is a Lie algebra homomorphism. Indeed, since the free Lie algebra only includes relations that are present in every Lie algebra, one can show that there is a surjective homomorphism from a free Lie algebra onto any finitely generated Lie algebra.
Another example that just occurred to me: any associative ring can be thought of as a Lie algebra using the bracket $[a,b]=ab-ba$. Any ring homomorphism will induce a Lie algebra homomorphism.
