The A-like matrices for a hypercube

Let $D$ denote a positive integer and let $Q_D$ denote the graph of the $D$-dimensional hypercube. Let $X$ denote the vertex set of $Q_D$ and let $A \in \MX$denotetheadjacencymatrixof$Q_D$\mathrm{.}Amatrix$B \in \MX$iscalled$A$-\text{em}likewheneverboth(i)$BA = AB$;(ii)forall$x,y \in X$ that are not equal or adjacent, the $(x,y)$-entry of $B$ is zero. Let $\text{Al}$ denote the subspace of $\text{MX}$ consisting of the $A$-like elements. We decompose $\text{Al}$ into the direct sum of its symmetric part and antisymmetric part. We give a basis for each part. The dimensions of the symmetric part and antisymmetric part are $D+1$ and $\left(\genfrac{}{}{0px}{}{D}{2}\right)$, respectively.