Complexification of a real Lie algebra

Let ${\mathfrak{g}}_0$ be a finite-dimensional real Lie algebra (Lie algebras over a field). Its complexification is the real tensor product ${\mathfrak{g}}_{\mathbb{C}}={\mathfrak{g}}_0{\otimes }_{\mathbb{R}}\mathbb{C}$ (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums), equipped with the scalar multiplication $t(X\otimes z)=X\otimes (tz)$ and with the unique complex-bilinear Lie bracket extending the original one, that is

$[X\otimes z,Y\otimes w]=[X,Y]\otimes zw(X,Y\in {\mathfrak{g}}_0,z,w\in \mathbb{C}),$

extended to all of ${\mathfrak{g}}_{\mathbb{C}}$ by linearity. Well-definedness and the Lie-algebra axioms can be checked without using any later property of the complexification. Indeed, the tensor-product relations give the canonical real-linear isomorphism $\mathrm{\Phi }:{\mathfrak{g}}_0{\otimes }_{\mathbb{R}}\mathbb{C}⟶{\mathfrak{g}}_0\oplus {\mathfrak{g}}_0,\mathrm{\Phi }(X\otimes (a+ib))=(aX,bX),$ whose inverse is $(U,V)\mapsto U\otimes 1+V\otimes i$. Under $\mathrm{\Phi }$ the displayed bracket is the unambiguous formula $[(X,Y),(U,V)]=([X,U]-[Y,V],[X,V]+[Y,U])\mathrm{.}$ It is complex bilinear for $i(X,Y)=(-Y,X)$; skew-symmetry and the Jacobi identity follow componentwise by expanding and using those identities in ${\mathfrak{g}}_0$. Thus the formula defines a complex Lie algebra directly.

The map $\epsilon \text{\hspace{0.17em}⁣}:{\mathfrak{g}}_0\to {\mathfrak{g}}_{\mathbb{C}}$, $\epsilon (X)=X\otimes 1$, is the canonical real embedding, and we write ${\mathfrak{g}}_0^{\mathbb{C}}:={\mathfrak{g}}_{\mathbb{C}}$. Every element of ${\mathfrak{g}}_{\mathbb{C}}$ has a unique expression $X\otimes 1+iY\otimes 1$ with $X,Y\in {\mathfrak{g}}_0$; the real part of such an element is $X$ and its imaginary part is $Y$.