Exercise 25, § 1

The formal linear group in n variables over the field K, denoted by $\mathbf{GL}(n, K)$ (when no confusion can arise), is the formal group of dimension $n^2$

whose group law is defined by $f_{ij}(u, v) = -\delta_{ij} + \sum_{k=1}^n (\delta_{ik} + U_{ik})(\delta_{kj} + V_{kj})$ ($\delta_{ij}$ the Kronecker index; the $3n^2$ indeterminates of the general definition of Exercise 24 are here denoted by $U_{ij}, V_{ij}, W_{ij}$ with 2 indices varying from 1 to n). Show that, if $D_{ij} = \partial / \partial U_{ij}$, the left invariant derivations $X_{ij}$ such that $X_{ij}^{(e)} = D_{ij}^{(e)}$ are given by

The Lie algebra of $\mathbf{GL}(n, K)$ is identified with $\mathfrak{gl}(n, K)$ by identifying $X_{ij}$ with the element $E_{ij}$ of the canonical basis (use formula (3) of Exercise 24). If K is of characteristic $p > 0$, $X_{ii}^p = X_{ii}$ and $X_{ij}^p = 0$ for $i \neq j$.