Beautiful MatrixPlot’s of E8 and H4.

This is not the typical E8 Gram matrix representation as the Cartan matrix using only the simple roots (shown at the end of this post).

It uses all 240 E8 vertices (+120/-120 roots) of the E8 Split Real Even (SRE) vertices to produce the Gram matrix (with vertices sorted in simple numeric order):

The same 240 E8 vertices (+120/-120 roots) below are sorted by the 9th row of the binary structure Pascal Triangle, excluding the 2nd & 2nd to last entries of -/+ 8 (i.e. {-1,-8,-28,-56,70=(-35/35),56,28,8,1}, where the 112 integer D8 are -56/56 and the 128 half-integer BC8 are {-1,-28,-35,35,28,1}).

As above, but with H4+H4φ 8D Gram matrix created using the dot product of E8 against the 8×8 E8 to H4 folding matrix U:

Now with only the H4φ 4D Gram matrix created as 120 H4 4D vertices:

Compare this to the Gram matrix using the root weights of E8 (in Pascal triangle order vs. sorted into root and/or weight order):

Then compare to the Cartan matrix produced as the Gram matrix of only the 8 E8 simple roots: