Doubt in understading Proof of Matrix lie group and Lie algebra locally homemorphic

Question

I was reading Brian C Hall Lie Group book In that I encountered following proof . I understand Whole proof. But have one doubt Why Auther take Orthogonal complement into consideration As I think Direct sum also be ok.

I thought hard But I do not understand why Author requires orthogonal complement.

I uploaded photo Because I do not want to miss any details ..

I would be thankful if someone helps me

Any help will be appreciated

Answer 1

The "orthogonal" part of that isn't important. What's crucial is that it's a complementary space, so that $\mathfrak{g}\oplus D$ is exactly the full space $M_n(\mathbb{C})$. We use this $D$ to extend the exponential map from $\mathfrak{g}$ to the full space - but not as the standard matrix exponential (We only have $e^{X+Y}=e^Xe^Y$ if $X$ and $Y$ commute). We use the inverse of this extended map, and that's why we needed a complementary space.

Why use the (real) orthogonal complement? Because it's a simple complementary space we can just write down and move on from.