A Remarkable New Identity Satisfied by the Dirac Matrices of a Bilocal Field Theory

In 1925 Elie Cartan described `triality’ \cite{CARTAN25}, \cite{CARTAN} as a symmetry between SO$(8;\mathbb{C})$ vectors and the two types of Spin$(8; $\mathbb{C}$)$ spinor. It is known that the reduced generators of the Clifford algebra ${\mathbb{C}}_8$ defined on the real, eight-dimensional Euclidean space ${\mathbb{E}}_8$ satisfy an identity that guarantees the existence of matrix representations (acting on the vector and spinor bundles of ${\mathbb{E}}_8$) of triality. Analogously, let ${\mathbb{E}}_{4,4}$ denote a real eight-dimensional pseudo-Euclidean vector space that is endowed with an indefinite inner product with signature $(+,+,+,-;-,-,-,+)$. As a normed vector space, ${\mathbb{E}}_{4,4}\cong M_{3,1}\times {}^{\ast }\text{\hspace{0.17em}⁣}{M}_{3,1}$, where $M_{3,1}$ and ${}^{\ast }\text{\hspace{0.17em}⁣}{M}_{3,1}$ denote real four-dimensional Minkowski spacetimes, with opposite signatures. %Clearly, bilocal Minkowski field theories may be cast on the ${\mathbb{E}}_{4,4}$ spacetime. The reduced generators (i.e., the Dirac matrices) of the pseudo Clifford algebra ${\mathbb{C}}_{4,4}$ defined on ${\mathbb{E}}_{4,4}$ satisfy an identity $$ \cite{NASH86} $,$ \cite{NASH90} that guarantees the existence of invertible linear mappings between each of the two types of $\overline{S0(4,4;\mathbb{R})}$ spinor and the ${S0(4,4; $\mathbb{R}$)}$ vector, thereby realizing matrix representations of triality that act on the vector and spinor bundles of the spacetime ${\mathbb{E}}_{4,4}$. In this note we generalize this identity (see Eq.[\ref{newIdentity}]).