Is a Mapping Between Lie Algebras an Isomorphism if it Takes a Basis to a Basis?

[Ted123]

If a mapping between Lie algebras $\varphi : \mathfrak{g} \to \mathfrak{h}$ takes a basis in $\mathfrak{g}$ to a basis in $\mathfrak{h}$ is it an isomorphism of vector spaces?

 

[Office_Shredder]

Good question. What do you think?

 

[Ted123]

Good question. What do you think?

I'm fairly sure it is. Is that right?

 

[HallsofIvy]

Assuming that by "takes a basis to a basis" you mean "one to one and onto", a Lie Algebra is completely determined by its basis, isn't it?