Exercise 2, § 11

Tag 017U, permanent. Cite it as [Bourbaki, Tag 017U].

Lie Groups and Lie Algebras, VIII, 262

Let

A=\begin{pmatrix}0&1\\0&0\end{pmatrix},\quad B=\begin{pmatrix}0&0\\1&0\end{pmatrix}.

Then

A

is nilpotent,

AB

is not, and

[A,[A,B]]=\begin{pmatrix}0&-2\\0&0\end{pmatrix}.

Deduce that Lemma 5 does not extend to fields of characteristic 2.

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