Covariance identity for continuous square-integrable martingales ($\mathbb{E}[M_t{N}_t]=\mathbb{E}[M,N{]}_t$)

For two zero-starting continuous martingales square-integrable at time $t$, the Itô isometry gives $\mathbb{E}(M_t\pm N_t{)}^2=\mathbb{E}[M\pm N{]}_t$. The polarization identity yields the displayed identity. Integrability of the cross-bracket follows from the Kunita-Watanabe inequality and the Cauchy-Schwarz inequality.