Weakly orthogonal continuous martingales ($\mathbb{E}[M_s{N}_t]=0(s,t\ge 0)$)

For zero-starting L2-bounded continuous martingales, this condition is equivalent to $\mathbb{E}[M_t{N}_t]=0$ at equal times, by conditioning the later factor. The covariance identity for continuous square-integrable martingales also makes it equivalent to $\mathbb{E}[M,N{]}_t=0$. It is weaker than the vanishing bracket condition for orthogonal continuous local martingales: $B_{t\wedge 1}$ and $B_{t\wedge 1}^2-(t\wedge 1)$ have zero equal-time covariance but nonzero bracket $2{\int }_0^{t\wedge 1}{B}_{s}ds$.