Remark

When $k$ is $\mathbf{R}$ or $\mathbf{C}$ or a complete non-discrete ultrametric field, $\mathfrak{s}\mathfrak{l}(2, k)$ is the Lie algebra of $\mathbf{S}\mathbf{L}(2, k)$. Let $\rho$ and $\pi$ be as in Th. 2. The homomorphism $\pi$ is a homomorphism of Lie groups from $\mathbf{S}\mathbf{L}(2, k)$ to $\mathbf{G}\mathbf{L}(E):$ this is clear when $E = V(m)$, and the general case follows, in view of Th. 1. By Chap. VII, §3, no. $1,\rho (X_+) = L(\pi )(X_+),\rho (X_-) = L(\pi )(X_-)$. Hence $\rho = L(\pi )$ (for a converse, see Exerc. 18).