Summary

Source code

This is constructed from VisibLie_E8 found on TheoryOfEverything.org archive copy at the Wayback Machine with the following settings:
(* This is an auto generated list from e8Flyer.nb *)

new := { 
scale=0.06;
cylR=0.021;
showEdges=True;
eGrad=0;
edgeVal={1, 12};
edgeDimTrim=3;
vOverlapColor=True;
pListName="C3=3Cube";
newPTcd={
{1/Sqrt[3], 1/Sqrt[3], 1/Sqrt[3], 0, 0, 0, 0, 0},
{-(1/Sqrt[2]), 0, 1/Sqrt[2], 0, 0, 0, 0, 0},
{-(1/Sqrt[2]), 0, 1/Sqrt[2], 0, 0, 0, 0, 0}};
};new;

Credit: User_talk:Tomruen for sharing basis vectors resulting in my code for n-cube Pascal projections:

doCubePascal@n_ := (
H = Normalize@Join[Array[1 &, n],Array[0 &, 8 - n]];
V = Normalize@Join[Table[i, {i, -(n - 1), n - 1, 2}],Array[0 &, 8 - n]];)

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