Example

Tag 0A3G, permanent. Cite it as [Bourbaki, Tag 0A3G].

Lie Groups and Lie Algebras, chapter II, § 2. Free Lie algebras

Suppose that $X$ has two elements $x, y$ ; let $X$ be ordered such that $x < y$ . The construction given in the proof of Proposition 11 gives a set $H$ with 14 elements of length $\leq 5$ given in the following table:

\begin{array}{cccc} H_1 & w_1 = x \\ H_2 & w_3 = (xy) \\ H_3 & w_4 = (x(xy)) & w_5 = (y(xy)) \\ H_4 & w_6 = (x(x(xy)))) & w_7 = (y(x(xy)))) & w_8 = (y(y(xy)))) \\ H_5 & w_9 = (x(x(x(xy)))) & w_{10} = (y(x(x(xy)))) & w_{11} = (y(y(x(xy))))) \\ & w_{12} = (y(y(y(xy)))) & w_{13} = ((xy)(x(xy))) & w_{14} = ((xy)(y(xy))). \end{array}

(The elements of $H$ have been numbered according to the total ordering chosen on each $H_n$ .)

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

Cited by

Nothing.

Cites

Proposition 11