Three historical(reverse order) forms of the E8<->(Subscript[H, 4]) folding matrix:

The current centrosymmetric, traceless (Tr=0), and volume preserving (Det = 1) form with features analyzed in:

The_Isomorphism_of_H4_and_E8 https://arxiv.org/abs/2311.01486

The_Isomorphism_of_3-Qubit_Hadamards_and_E8  https://arxiv.org/abs/2311.11918

It transforms a (Left 4D) (Subscript[H, 4]) (the 120 vertex 600-cell) into a second scaled (palindromically reversed Right 4D) φ(Subscript[H, 4]) copy with a Galois transform φ↔-1/φ. 
Those L+R (now 8D) are palindromically reversed and combined into 240 vertices by (Subscript[H, 4]^2)=(Subscript[H, 4 L+R])+φ(Subscript[H, 4 L+R]) that give you back (Subscript[E, 8]) upon rotation (folding/unfolding) by (U^(-1)).

matCheck@octSym@Det1𝕌//MatrixForm

inN@Det@Det1𝕌

Det1𝕌=((Det1𝕌)^(-1)) by swapping (non-outside corner)(Sqrt[φ])↔(1/(Sqrt[φ]))

octSym@Inverse@Det1𝕌//MatrixForm

((Det1𝕌)^2) as the Dot product forms diagonal Cartan-like matrix of this folding matrix  Det1𝕌.Det1𝕌:

matCheck@octSym[Det1𝕌.Det1𝕌]//MatrixForm

An older (but easier to read symbolically) Det≠1 form:

This is just an unscaled version of the current Det1𝕌...

𝕌Det1f
Det1𝕌==𝕌/%//MatrixForm

matCheck@octSym@𝕌//MatrixForm

inN@Det@𝕌

octSym@Inverse@𝕌//MatrixForm

(𝕌^2) as the Dot product forms diagonal Cartan-like matrix of this folding matrix  𝕌.𝕌:

matCheck@octSym[𝕌.𝕌]//MatrixForm

The original 2012-2014 L/R quadrant preserving as an (Subscript[E, 8]) to canonical 600-cell ((Subscript[H, 4]) family of polytopes)  form :

Where L/R quadrants have a Galois transform of ϕ ↔ -ϕ with upper left corners exchanging Φ↔(ϕ^2):

H4fold=(({{Φ, 0, 0, 0, (ϕ^2), 0, 0, 0}, {0, ϕ, 1, 0, 0, -ϕ, 1, 0}, {0, 1, 0, ϕ, 0, 1, 0, -ϕ}, {0, 0, ϕ, 1, 0, 0, -ϕ, 1}, {(ϕ^2), 0, 0, 0, Φ, 0, 0, 0}, {0, -ϕ, 1, 0, 0, ϕ, 1, 0}, {0, 1, 0, -ϕ, 0, 1, 0, ϕ}, {0, 0, -ϕ, 1, 0, 0, ϕ, 1}}));

matCheck@H4fold

inN@Det@H4fold

octSym@Inverse@H4fold//MatrixForm

(H4fold^2) as the Dot product forms diagonal Cartan-like matrix of this folding matrix  H4fold.H4fold:

matCheck@octSym[H4fold.H4fold]//MatrixForm