Cohomology of complex Lie groups via compact form

Question

Let $G$ be a compact Lie group. Let $G_{\mathbb{C}}$ be a complex Lie group such that there is inclusion $i: G \rightarrow G_{\mathbb{C}}$ of Lie groups. Moreover I require that differential of $i$ induces isomorphism $Lie(G) \otimes \mathbb{C} \rightarrow Lie( G_{\mathbb{C}} )$.

Question Is $i$ homotopy equivalence?

Remark Basic example is $G = SU_n$ and $G_{\mathbb{C}} = SL_n ( \mathbb{C} )$.

Answer 1

Let $S^1$ be the quotient of $R$ by $(x\rightarrow x+1)$ and $T^2$ the quotient of $C$ by $(x\rightarrow x+1, x\rightarrow x+i)$ you have an inclusion $i:S^1\rightarrow T^2$ induced by the canonical inclusion $R\rightarrow C$ such that $Lie(S^1)\otimes C=Lie(T^2)$ but $i$ is not an homotopic equivalence.