Consider the following model,
\[\begin{aligned} \bm y \mid \bm\beta &\sim \MvNormal(\bm X \bm \beta, \bm\Sigma) \\ \bm \beta & \sim \MvNormal(\bm m, \bm S) \end{aligned}\]Using section 14 from BDA3, we can write the above equivalently as:
\[\begin{aligned} \begin{pmatrix} \bm \Sigma ^{-1/2} \bm y \\ \bm m \end{pmatrix} \Bigg| \bm \beta &\sim \MvNormal\p{ \begin{bmatrix} \bm\Sigma^{-1/2} \bm X \\ \bm I \end{bmatrix} \bm \beta, \begin{bmatrix} \bm I & \bm 0 \\ \bm 0 & \bm S \end{bmatrix} }. \end{aligned}\]Thus, we obtain the following posterior for $\bm\beta$:
\[\begin{aligned} \bm \beta \mid \bm y &\sim \MvNormal(\bm m^\star, \bm S^\star), \text{ where} \\ \bm S^\star &= \p{\bm X^T \bm\Sigma^{-1} \bm X + \bm S^{-1}}^{-1} \\ \bm m^\star &= \bm S^\star \p{\bm X^T \bm\Sigma^{-1} \bm y + \bm S^{-1} \bm m} \\ \end{aligned}\]