Weyl's unitary trick

Remarks

Example: for $\mathfrak{g} = \mathfrak{sl}(2, \C)$, one takes $\mathfrak{k} = \mathfrak{su}(2)$ and $K = SU(2)$. Complete reducibility of representations of $SU(2)$, proved by averaging an inner product, gives complete reducibility of finite-dimensional representations of $\mathfrak{sl}(2, \C)$. Non-example: the real form $\mathfrak{sl}(2, \R)$ of $\mathfrak{sl}(2,\C)$ cannot replace $\mathfrak{k}$ in step 3, because every connected Lie group with Lie algebra $\mathfrak{sl}(2,\R)$, such as $SL(2,\R)$, is not compact. The same method applies to a connected complex semisimple Lie group $G$ with a compact real form $K$, using holomorphic representations of $G$. In step 3, the integrated representation of $K$ has the same invariant subspaces as the representation of $\mathfrak{k}$. The compactness of $K$ in step 3 is a theorem of Weyl. It says that a connected Lie group whose Lie algebra has a negative definite Killing form is compact, and so is its universal cover.