On the complexification of the commutator ideal

Question

Let $\mathfrak{g}$ be a real Lie algebra and $\mathfrak{g}_{\mathbb{C}}$ its complexification. Then, since $\mathfrak{g}$ is a Lie subalgebra of $\mathfrak{g}_{\mathbb{C}}$, the commutator ideal $[\mathfrak{g},\mathfrak{g}]$ is contained in $[\mathfrak{g}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}}]$. Now the complexification of $[\mathfrak{g},\mathfrak{g}]$ is $[\mathfrak{g},\mathfrak{g}]_{\mathbb{C}}$ and is obviously contained in $\mathfrak{g}_{\mathbb{C}}$ (since the complexification of a subspace must be contained in the complexification of the space itself).

My question is: Why does then follow that $[\mathfrak{g},\mathfrak{g}]_{\mathbb{C}} \subset [\mathfrak{g}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}}]$? We only know that $[\mathfrak{g}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}}] \subset \mathfrak{g}_{\mathbb{C}}$ and from that we can't conclude the claim, or can we? It is probably utterly trivial but I just don't see it.

Answer 1

The complexification is given by $$ _ℂ=⊗_ℝℂ. $$ Then we have $$ [\mathfrak{g}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}}]=[\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C},\mathfrak{g}\otimes_{\mathbb{R}}\mathbb{C}]=[\mathfrak{g},\mathfrak{g}]\otimes_{\mathbb{R}}\mathbb{C}=[\mathfrak{g},\mathfrak{g}]_{\mathbb{C}}, $$ and the commutator is given by $$ [x_1+iy_1,x_2+iy_2]=[x_1,x_2]-[y_1,y_2]+i([x_1,y_2]+[y_1,x_2]. $$