# $\left(n+1\right)\times \left(n+1\right)$ algebra isomorphic to Bose-Mesner algebra?

The Wikipedia article on association schemes claims regarding Bose-Mesner algebras:

There is another algebra of $\left(n+1\right)\times \left(n+1\right)$ matrices which is isomorphic to ${\mathcal {A}}$, and is often easier to work with.

What is this algebra called? In which paper(s) is it defined and used? Unhelpfully, the statement is not cited.

Let $\mathcal{A}$ be the Bose-Mesner algebra of an association scheme $(X,\mathcal{R})$ with $n$ classes and with intersection numbers $p_{ij}^k$. Let $D_0,D_1,\dots,D_n$ be the adjacency matrices that form a basis for the Bose-Mesner algebra. For $i=0,1,\dots,n$, let $L_i$ be an $(n+1)\times(n+1)$ matrix with $L_i=(p_{ij}^k)_{0\leq j,k\leq n}$. The $\mathbb{C}$-vector space $\mathcal{L}:=\langle L_0,L_1,\dots,L_n\rangle$ is a matrix subalgebra of $M_{n+1}(\mathbb{C})$. One way to prove this is by using the associativity of matrix multiplication: $D_i(D_jD_k)=(D_iD_j)D_k$. Moreover, we get an algebra isomorphism by sending $D_i\mapsto L_i$. This can be shown by using some properties of the intersection numbers. Hence, the algebra $\mathcal{L}$ is isomorphic to the Bose-Mesner algebra. In particular, the matrix $L_i$ has the same eigenvalues as the adjacency matrix $D_i$. This can be helpful if $|X|$ is large compared to $n$ since $D_i$ is an $|X|\times |X|$ matrix and $L_i$ is an $(n+1)\times(n+1)$ matrix.
If you look at this from a representation theoretical point of view, then the matrices $L_i$ correspond to the regular representation of $\mathcal{A}$.