Proposition 17

Let $K$ be a commutative field. The unimodular group $\mathbf{SL}(n, K)$ is generated by the matrices $B_{ij}(\lambda)$ with $i \neq j$ and $\lambda \in K$.

By II, § 10, no. 13, Proposition 14, we know that every matrix in $\mathbf{GL}(n, K)$ is a product of matrices of the form $B_{ij}(\lambda)$ and a matrix of the form $\operatorname{diag}(1, 1, \ldots, 1, \alpha)$ with $\alpha \in K^*$. Now it is immediate that $\det(B_{ij}(\lambda)) = 1$ and $\det(\operatorname{diag}(1, \ldots, 1, \alpha)) = \alpha$ (no. 6, Example 2); whence the proposition.