Exercise 4, § 2

Lie Groups and Lie Algebras, II, 181

(a) Show that the Lie algebra defined by the presentation

\{x, y; [x, [x, y]] = [y, [x, y]] = 0\}

admits the basis $(x, y, [x, y])$ . Define an embedding of this algebra in a matrix algebra.

(b) Do the same for the Lie algebra defined by the presentation

\{x, y; [x, [x, y]] - 2x = [y, [x, y]] + 2y = 0\}.

\{x, y, z; [x, y] - x = [y, z] - y = [z, x] - z = 0\}

reduces to 0.

Solution

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