I am reading a book on linear regression and have some trouble understanding the variance-covariance matrix of $\mathbf{b}$:

The diagonal items are easy enough, but the off-diagonal ones are a bit more difficult, what puzzles me is that $$ \sigma(b_0, b_1) = E(b_0 b_1) - E(b_0)E(b_1) = E(b_0 b_1) - \beta_0 \beta_1 $$

but there is no trace of $\beta_0$ and $\beta_1$ here.