Corollary 2

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

Lie Groups and Lie Algebras, chapter IV, Appendix 0. Graphs

A subset $S'$ of $S$ is a union of connected components if and only if no vertex belonging to $S'$ is joined to any vertex belonging to $S - S'$ .

The condition is sufficient by Prop. 1, (iv). It is necessary by Prop. 1, (iii).

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

Cited by

Nothing.