The based loop group, $\Omega G$, is known to admit a Kaehler metric, given as \begin{equation} g(X,Y)=2\sum_{k>0}k\textrm{Tr}(X_{-k}Y_k), \end{equation} this is given in page 150 of Segal and Pressley's 'Loop Groups', where $X,Y\in L\mathfrak{g}$, the loop algebra. $\Omega G$ can be described as the homogenous space $LG/G$, and there is a transitive $LG$ isometry on $\Omega G$.
It is known that the above metric can be shown to be $LG$-invariant, see e.g., Khesin and Wendt's 'The Geometry of Infinite Dimensional Groups', page 239, below Corollary 4.8, and Armen Sergeev's 'Kahler Geometry of Loop Spaces', page 100.
How does one show the $LG$ invariance explicitly? In other words, what are the correct transformations of $X$ and $Y$ which would leave the metric invariant? The Killing form for a Loop Algebra is invariant under adjoint transformations, so I would expect something similar. An additional reference which is useful is Dan Freed's 'The Geometry of Loop Groups'.