Remark

(1) $\mathbf{GL}(2, \mathbf{F}_2) = \mathbf{SL}(2, \mathbf{F}_2)$; this is a solvable group of order 6, whose commutator group is of index 2 (II, § 10, Exercise 14).

(2) With the same notation as above, it can be proved as in I, § 5, no. 7, Proposition 9 that, for $i < j, j - i > 1$, $u_{ij} = u_{j-1, j} u_i, _{j-1} u_{j-1, j}^{-1}$; hence the group $\mathbf{SL}(n, K)$ is generated by the matrices $B_{12}(\lambda)$ and $U_{i, i+1}$ for $1 \leq i \leq n - 1$.