Exercise 1, § 11

Let $\mathfrak{g}=\mathfrak{s}\mathfrak{l}(2, k)$. Put G = Aut$_e(\mathfrak{g})$; this group can be identified with $\mathbf{P}\mathbf{S}\mathbf{L}_2(k)$, cf. Chap. VII, §3, no. 1, Remark 2.

a) Every nilpotent element of $\mathfrak{g}$ is G-conjugate to $(0$0 $\lambda 0)$ for some $\lambda \in k$.

Such an element is principal if and only if it is non-zero.

$^{19}$In fact, (iii) is equivalent to (i) and (ii). For this, see: R. STEINBERG, Regular

elements of semi-simple algebraic groups, Publ. Math. I.H.E.S., Vol. XXV (1965),

pp. 49-80. b) The elements

where $\lambda,\mu\in k^*$, are G-conjugate if and only if $\lambda^{-1}\mu$ is a square in $k$.

c) Every simple element of $\mathfrak{g}$ is G-conjugate to