Proposition 4

Tag 0A1W, permanent. Cite it as [Bourbaki, Tag 0A1W].

The set $P(E)$ of primitive elements of $E$ is a Lie subalgebra of $E$ .

If $x, y$ are in $P(E)$ , then

\begin{align*} c(xy) &= c(x)c(y) = (x \otimes 1 + 1 \otimes x)(y \otimes 1 + 1 \otimes y) \\ &= xy \otimes 1 + 1 \otimes xy + x \otimes y + y \otimes x, \end{align*}

whence

c([x, y]) = [x, y] \otimes 1 + 1 \otimes [x, y].

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

Cited by

Nothing.