Let $k$ be a number field and let $G$ be a quasisplit reductive algebraic group over $k$. Does there exist a maximal torus in $G$ such that the Hasse principle in dimension $2$ holds, i.e., such that the map $H^2(k,T) \to \prod_v H^2(k_v,T)$ is injective? It is true if $G$ is split or if $G$ is adjoint. If $G$ is simply connected, it is true whether $G$ is quasisplit or not.

Yes, if $G$ is semisimple, not necessarily quasisplit. Indeed, let $v$ be a nonarachimedean place of $k$. Then there exists an anisotropic maximal $k_v$-torus $T_v\subset G_{k_v}$; see Platonov and Rapinchuk, Section 6.5, Theorem 21 of the Russian edition. There exist a maximal $k$-torus $T\subset G$ that is conjugate to $T_v$ over $k_v$. Thus $T_{k_v}$ is anisotropic. It follows that $Ш^2(k,T)=1$; see formula (1.9.3) in Sansuc's paper: Sansuc, J.-J. Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math. 327 (1981), 12–80. $\endgroup$2more comments