a maximal torus $T$ of $G_{k^{{\operatorname{sep}}}}$, the group $X^*$ is isomorphic to the character group of $T$ but not canonically; however given also a Borel $B$ containing the torus, there is now a canonical map $X^*=X^*(T)$ (and different Borels give different canonical isomorphisms). There is a natural group homomorphism $\mu_G:\Gamma_k\to{\operatorname{Aut}}(\Psi_0(G))$ (defined for example in §1.3 of [@MR546608]) and if $K\subseteq k^{{\operatorname{sep}}}$ is a Galois extension of $k$ that splits $G$ then $\mu_G$ factors through ${\operatorname{Gal}}(K/k)$.

We let ${{\widehat{G}}}$ denote a connected reductive group over ${\overline{{\mathbb{Q}}}}$ equipped with a given isomorphism $\Psi_0({{\widehat{G}}})=\Psi_0(G)^\vee$, the dual root datum to $\Psi_0(G)$. There is a canonical group isomorphism ${\operatorname{Aut}}(\Psi_0(G))={\operatorname{Aut}}(\Psi_0(G)^\vee)$, sending an automorphism of $X^*$ to its inverse (one needs to insert this inverse to ’cancel out’ the contravariance coming from the dual construction), and hence a canonical action of $\Gamma_k$ on $\Psi_0(G)^\vee$. If we choose a Borel, a maximal torus, and a splitting (also called a pinning; see p.10 of [@MR546587] for details and definitions) of ${{\widehat{G}}}$ then, as on p.10 of [@MR546587], this data induces a lifting ${\operatorname{Aut}}(\Psi_0(G)^\vee)\to{\operatorname{Aut}}({{\widehat{G}}})$ and hence (via $\mu_G)$ a left action of $\Gamma_k$ on ${{\widehat{G}}}$. We define the $L$-group ${{{}^{L}G}}$ of $G$ to be the resulting semidirect product, regarded as 