Example

Tag 0A0C, permanent. Cite it as [Bourbaki, Tag 0A0C].

Lie Groups and Lie Algebras, chapter I, § 6. Semi-simple Lie algebras

We identify $\mathrm{sl}(K^2)$ with the Lie algebra of matrices of order 2 and trace zero. We write

X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \quad Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \quad H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

Then $X, Y, H$ form a basis of $\mathrm{sl}(K^2)$ and

[H, X] = 2X \qquad [H, Y] = -2Y \qquad [X, Y] = H.

As an algebra of dimension 1 or 2 is not semi-simple (no. 1, Remark 1), $\mathrm{sl}(K^2)$ is simple. In fact, $\mathrm{sl}(V)$ is simple for $\dim V \geq 2$ , as we shall see later (cf. also Exercises 21 and 24).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 9, § 6

Cites

Remark 1