Reference for "how to read of the faces of a uniform polytope from its Coxeter diagram"

Question

It appears as if the (combinatorial type of the) faces of (Wythoffian) uniform polytopes can be read off from their Coxeter-Dynkin diagram by deleting appropriate vertices. I think I understood how this is done. But ...

Question: Where in the literature is this proven rigorously?

Please be more specific than "Coxeter's book on uniform polytopes".

Answer 1

Maxwell's 1989 paper I think addresses it but it requires a bit of work to decipher https://core.ac.uk/download/pdf/82287983.pdf

See also Scharlau https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-61.3.615

Answer 2

Restricting your quest to regular polytopes, then indeed Coxeter's book on Regular Polytopes is the searched for source: Within §7.5 on p.129 he states

Since the cells of the vertex figure are vertex figures of cells, a regular $\Pi_4$ whose cells are $\{p,q\}$ must have vertex figures $\{q,r\}$. (Here $r$ is simply the number of cells that surround an edge.) Accordingly we write $$\Pi_4=\{p,q,r\}\ ;$$ e.g., $$\alpha_4=\{3,3,3\},\ \beta_4=\{3,3,4\},\ \gamma_4=\{4,3,3\}$$ (cf. 7.27). Similarily, a regular $\Pi_5$ whose cells are $\{p,q,r\}$ must have vertex figures $\{q,r,s\}$, and we write $$\Pi_5=\{p,q,r,s\}.$$

The remainder here is only to translate the notation by Schläfli symbols into those by Coxeter-Dynkin diagrams.

Thus, i.e. by mere construction, the facets are derived from a given diagram of a regular polytope by deletion of the farest unringed node, while the vertex figure of that polytope is obtained by deleting the ringed node and ringing the formerly connected one instead.

By means of Wythoff's kaleidoscopical construction the analogue deletion of any node of the diagram of a (Wythoffian) uniform polytope then follows.

--- rk