  In this paper we consider a distance-regular graph $\Gamma$. Fix a vertex $x$
of $\Gamma$ and consider the corresponding subconstituent algebra $T$. The
algebra $T$ is the $\mathbb{C}$-algebra generated by the Bose-Mesner algebra
$M$ of $\Gamma$ and the dual Bose-Mesner algebra $M^*$ of $\Gamma$ with respect
to $x$. We consider the subspaces $M, M^*, MM^*, M^*M, MM^*M, M^*MM^*, \dots$
along with their intersections and sums. In our notation, $MM^*$ means
$Span\{RS|R\in M, S\in M^*\}$, and so on. We introduce a diagram that describes
how these subspaces are related. We describe in detail that part of the diagram
up to $MM^*+M^*M$. For each subspace $U$ shown in this part of the diagram, we
display an orthogonal basis for $U$ along with the dimension of $U$. For an
edge $U\subseteq W$ from this part of the diagram, we display an orthogonal
basis for the orthogonal complement of $U$ in $W$ along with the dimension of
this orthogonal complement.
