The algebraic Brauer group of a reductive group over a nonarchimedean local field

We show that for nonarchimedean local fields $F$, the pairing from the algebraic part of the Brauer group of a reductive group $G$ characterizes all continuous homomorphisms from $G(F)$ into $\mathbb{Q}\mathrm{/}\mathbb{Z}$. This generalizes results of Loughran and Loughran-Tanimoto-Takloo-Bighash.