Name for facet of a cone containing all but one edge

Let $ C \subseteq \mathbb R^n$ be a polyhedral cone, so generated by its edges ($ 1$ -dimensional faces) and $ F \subseteq C$ a facet (codimension $ 1$ face) of it containing every edge except $ e$ . In particular, the map $ F \times e \stackrel{+}{\to} C$ is an isomorphism of cones.

Is there a name for this property of $ F$ or $ e$ (which determine one another)?

Essentially, I regard the edges $ e$ like this as kind of trivial, and would like to split them off to deal with the difficult part of $ C$ . So maybe one could speak of a “core facet” $ F$ of $ C$ , and let the “core of $ C$ ” be the intersection of the “core facets”. Then the map $ core(C) \times \prod_{\text{core edges }e} e \stackrel{+}{\to} C$ would be an isomorphism, and $ core(C)$ would have no core edges.

There is a similar, familiar construction in the theory of simplicial complexes, where a “cone vertex” $ v$ of $ \Delta$ is one lying in every maximal face. One can safely delete all the cone vertices, and recone on them to reconstruct $ \Delta$ . Obviously one doesn’t want to steal this terminology directly and speak of “cone edges”.