# Reference request: Conditional posterior for bayesian hierarchical linear regression model

I am searching for a reference computing the conditional posterior distribution of a bayesian hierarchical linear regression model of the standard form: $$y\sim N(X\beta,\sigma I)\\ \sigma^2 \sim IG(v_1,v_2) \\ \beta|\mu,\sigma^2 \sim N(\mu,\Lambda_0 ^{-1})\\ \mu \sim N(\mu_0,\Lambda_1 ^{-1})$$ In my perception this model is a standard example on how to work with hierarchical models, however, I failed in finding one source that gives me the conditional posterior distributions such that I can set up a Gibbs sampler. Each and every snippet summarizing this computation is welcome, I would also be happy to create this computation in this post in order to make it available easier. (Source code of the Gibbs sampler is also appreciated.)