Covariance of summed Brownian motions

Let $B_1, B_2, B_3$ be independent standard Brownian motions. Compute $\operatorname{Cov}\big(B_1(1) + B_2(1),\ B_2(2) + B_3(2)\big)$ .

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Expand bilinearly; covariances between different (independent) Brownian motions are $0$ .
Only $B_2$ is shared: $\operatorname{Cov}(B_2(1),B_2(2))=\min(1,2)=1$ .

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