= Solution
The <stochastic process> $W$ starts from zero and has continuous paths. Its increments on disjoint intervals are <independent>, since the corresponding pairs of <Brownian motion> increments are <independent>, and
$$
W_t-W_s\sim N\bigl(0,(\rho^2+1-\rho^2)(t-s)\bigr)=N(0,t-s).
$$
Thus \b[$W$ is a standard <Brownian motion>, including $\rho=\pm1$.] Here the intended <filtration> is the joint natural <filtration>, or another <filtration> relative to which the pair is a two-dimensional <Brownian motion>; <independence> of their path laws alone would not justify arbitrary extra information in a larger <filtration>.
By bilinearity of <quadratic covariation> and part (i),
$$
\boxed{\langle W,B\rangle_t=\rho\langle B\rangle_t+\sqrt{1-\rho^2}\langle\widetilde B,B\rangle_t=\rho t.}
$$
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