Let $G$ be a semisimple Lie group. Its Killing form is a nondegenerate inner product on the tangent space to $G$ at the identity, and this form can be naturally extended to a metric on the whole of $G$ via left-translations.
One can show that the Levi-Civita connection on $G$ determined by the Killing form is $\nabla_X Y=\frac{1}{2}[X,Y]$ (where $X$ and $Y$ are left-invariant vector fields), using the Koszul formula. One can then show that the Riemann curvature is $R(X,Y)Z=\frac{1}{4}[[X,Y],Z]$.
Now, consider the complexification of $G$, denoted as $G_\mathbb{C}$. Can one define a Hermitian metric on $G_\mathbb{C}$ using the Killing form, i.e., by identifying a complex structure, $J$, on $G_\mathbb{C}$, and constructing the Hermitian metric as $h(X,Y)=\frac{1}{2}(g(X,Y)+g(JX,JY))$, where $g$ is the Killing form?
If yes, can one then go on to find the Levi-Civita connection and Riemann curvature in the form of commutators, in an analogous manner to how its done for $G$?