Exercise 7, § 4

Tag 011M, permanent. Cite it as [Bourbaki, Tag 011M].

Lie Groups and Lie Algebras, VIII, 233

Bourbaki marks this one as harder than the rest.

Let A be a commutative ring in which 2 is invertible, and let $\mathfrak{u}$ be the A-Lie algebra defined by the family of generators $\{x, y\}$ and the relations

[x,[x, y]] = 0,[y,[y,[y,[y, x]]]] = 0

Show that $\mathfrak{u}$ is a free A-module with basis

\{x, y,[x, y],[y,[x, y]],[y,[y,[x, y]]],[x,[y,[y,[x, y]]]]\}

Show that, when $A =k,\mathfrak{u}$ is isomorphic to the algebra $\mathfrak{a}_+/\mathfrak{n}$ corresponding to a root system of type $G_2$ .

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