lie algebras - Defining the Lie Bracket on $mathfrak{g} otimes_Bbb{R} Bbb{C}$

I already know how to do the complexification of a real Lie algebra $\mathfrak{g}$ by the usual process of taking $\mathfrak{g}_\Bbb{C}$ to be $\mathfrak{g} \oplus i\mathfrak{g}$. Now suppose I take the approach of trying to complexify things using tensor products. I look at $\mathfrak{g} \otimes_\Bbb{R} \Bbb{C}$ with the $\Bbb{R}$ - linear map

$$\begin{eqnarray*} f : &\mathfrak{g}& \longrightarrow \mathfrak{g} \otimes_\Bbb{R} \Bbb{C} \\ &v&\longmapsto v \otimes 1. \end{eqnarray*}$$

Now suppose I have an $\Bbb{R}$ - linear map map $h : \mathfrak{g} \to \mathfrak{h}$ where $\mathfrak{h}$ is any other complex Lie algebra. Then I can define a $\Bbb{C}$ - linear map $g$ from the complexification $\mathfrak{g} \otimes_\Bbb{R} \Bbb{C}$ to $\mathfrak{h}$ simply by defining the action on elementary tensors as

$$g(v \otimes i) = ih(v).$$

I have checked that $g$ is a $\Bbb{C}$ - linear map. Now my problem comes now in that my $f,g,h$ have to somehow be compatible with the bracket on $[\cdot,\cdot]_\mathfrak{g}$ of $\mathfrak{g}$ and $[\cdot,\cdot]_\mathfrak{h}$ of $\mathfrak{h}$. This is because I don't want them to just be linear maps but also Real/Complex Lie algebra homomorphisms.

My question is: How do we define the bracket on the complexification? A reasonable guess would be $[v \otimes i,w \otimes i] = \left([v,w] \otimes [i,i]\right)$ but this is zero.

Edit: Perhaps I should add, in the usual way of defining the complexification, the bracket on $\mathfrak{g}$ extends uniquely to one on the complexification $\mathfrak{g} \oplus i\mathfrak{g}$. Should it not be the case now that my bracket on $\mathfrak{g}$ extends uniquely to one on the tensor product then?

Edit: How do we know that the Lie Bracket defined by MTurgeon is well-defined? Does it follow from the fact that we are tensoring vector spaces, and so there is one and only one way to represent a vector in here?

Answer

First of all, it seems the right extension is the following:

$$[v\otimes\lambda,w\otimes\mu]:=[v,w]\otimes\lambda\mu.$$

This satisfies bilinearity, and Jacobi's identity. However, how can we show that this is the unique extension of the Lie bracket? We have the following result (taken, for example, from Bump's Lie groups):

Proposition: If $V$ and $U$ are real vector spaces, any $\mathbb R$-bilinear map $V\times V\to U$ extends uniquely to a $\mathbb C$-bilinear map $V_{\mathbb C}\times V_{\mathbb C}\to U_{\mathbb C}$.

Proof: This basically follows from the properties of tensor products. Any $\mathbb R$-bilinear map $V\times V\to U$ corresponds to a unique $\mathbb R$-linear map $V\otimes_{\mathbb R} V\to U$. But any $\mathbb R$-linear map extends uniquely to a $\mathbb C$-linear map of the complexified vector spaces (this is easy to prove). Hence, we have a $\mathbb C$-linear map $(V\otimes_{\mathbb R} V)_{\mathbb C}\to U_{\mathbb C}$. But we have the following isomorphism:

$$(V\otimes_{\mathbb R} V)_{\mathbb C}\cong V_{\mathbb C}\otimes_{\mathbb C} V_{\mathbb C};$$
on the left-hand side, the tensor product is over $\mathbb R$, and on the right-hand side, it is over $\mathbb C$. Finally, our $\mathbb C$-linear map $V_{\mathbb C}\otimes_{\mathbb C} V_{\mathbb C}\to U_{\mathbb C}$ corresponds to a unique $\mathbb C$-bilinear map $V_{\mathbb C}\times V_{\mathbb C}\to U_{\mathbb C}$.