  This paper presents several notable properties of the matrix $\mathbb{U}$
shown to be related to the isomorphism between $H_4$ and $E_8$. The most
significant of these properties is that $\mathbb{U}$.$\mathbb{U}$ is to rank 8
matrices what the golden ratio is to numbers. That is to say, the difference
between it and its inverse is the identity element, albeit with a twist.
Specifically, $\mathbb{U}$.$\mathbb{U}$-$ (\mathbb{U}$.$\mathbb{U})^{-1}$ is
the reverse identity matrix or standard involutory permutation matrix of rank
8. It has the same palindromic characteristic polynomial coefficients as the
normalized 3-qubit Hadamard matrix with 8-bit binary basis states, which is
known to be isomorphic to E8 through its (8,4) Hamming code.
