I am currently studying representation theory and the topic of discussion is complexifications of real Lie algebras.
I have noted down that the complexification $\mathfrak{g}^{c}$ of a real Lie algebra $\mathfrak{g}$ is the vector space $\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C}$ equppied with the bracket $[X\otimes z,Y \otimes w]^{c}=[X,Y]\otimes zw, X,Y\in\mathfrak{g}, z,w \in \mathbb{C}$.
How would one compute such a complexification explicity? What is let us say the complexification of the algebra $\mathfrak{gl}_{n}\mathbb{R}$, i.e. the set of matrices over $\mathbb{R}$?
An explicit computation would be greatly appreciated and very helpful in me grasping the concept.