Exercise 6, § 5

Lie Groups and Lie Algebras, I, 99

(a) Verify that the following multiplication table defines a 5-dimensional solvable Lie algebra $g$ :

[x_1, x_2] = x_5,\quad [x_1, x_3] = x_3,\quad [x_2, x_4] = x_4 \\ [x_1, x_4] = [x_2, x_3] = [x_3, x_4] = [x_5, g] = 0.

(b) Show that the orthogonal of $g$ with respect to the Killing form is $\mathscr{D}g = Kx_3 + Kx_4 + Kx_5$ . Deduce that $\mathscr{D}g$ is the largest nilpotent ideal of $g$ .

(c) Show that there exists no supplementary subalgebra of $\mathscr{D}g$ in $g$ . Deduce that $g$ is not the semi-direct product of a commutative algebra and a nilpotent ideal. (Show that this nilpotent ideal would necessarily be $\mathscr{D}g$ .)

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