Beautiful MatrixPlot’s of E8 and H4. The 240 E8 vertices (+120/-120 roots) 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 {1,38,35,-35,-28,-1}).

Below is the E8 Gram matrix of ALL roots (not the typical representation as the Cartan matrix using only the simple roots):

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

H4φ 4D Gram matrix created as 120 H4 4D vertices:

Compare this to the Cartan matrix produced as the Gram matrix of only the 8 E8 simple roots:

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