When does zero cross-quadratic variation imply independence of Brownian motions?

Question

We know that if $X$, $Y$ are Normal random variables with zero covariance then they are independent if and only if they are bivariate Normal, i.e. $aX+bY$ is Normal for all a,b. Similarly if $B_t$, $W_t$ are Brownian motions with zero cross-quadratic variation $\langle B,W\rangle_t$, and $aB_t+bW_t$ is Normal for all $a$, $b$ and $t$ then I can prove they are independent. Does anybody know of a weaker sufficient condition? I can show that $B_t$ and $cB_{\frac{t}{c^2}}$ have zero cross-quadratic variation for all $c\neq 1$ yet are dependent so I do need an extra condition. My advisor thinks that if the Brownian motions are adapted to the same filtration then that is sufficient but I'm not convinced and I can't find anything in the literature that explores this topic.