Block relative gain (BRG) is a useful method for finding suitable pairings for block decentralized control. In this paper, we present some new algebraic properties of BRG and establish its relation with closed-loop stability, controllability, block diagonal dominance, and interactions. Contrary to published conjecture, we show that strongly interacting systems can have BRG that is nearly the identity matrix. Our results provide simple rules for pairing of variables. We also extend the known method for calculating relative gain array for interacting systems to BRG.