# 2 Theorems and proofs

You can state and prove results, and refer to them, for instance Lemma 1 below.

Lemma 1

Splittings of $0\to \mathbf{G}_m\to \tilde{G}\to G\to 0$ canonically biject with twisting elements for $G$.

Proof

To give a splitting is to give a map $\tilde{G}\to \mathbf{G}_m$ such that the composite $\mathbf{G}_m\to {\tilde{G}}\to \mathbf{G}_m$ is the identity; then the induced map ${\tilde{G}}\to G\times \mathbf{G}_m$ is an injection with trivial kernel so is an isomorphism for dimension reasons. If $\chi :{\tilde{G}}\to \mathbf{G}_m$ is such a character then $\chi$ gives rise to an element of $X^*({\tilde{T}})$ which is Galois-stable, whose image in $\mathbf{Z}$ is 1, and which pairs to zero with each coroot (because $\chi$ factors through the maximal torus quotient of ${\tilde{G}}$). Conversely to give such a character is to give a splitting. Now one checks that $\theta -\chi$ has image in $\mathbf{Z}$ equal to zero so gives rise to an element of $X^*(T)$ which is Galois-stable, and pairs with each simple coroot to 1—but this is precisely a twisting element for $G$. Conversely if $t$ is a twisting element for $G$ then $\theta -t$ gives a splitting of the exact sequence.