Leonard pairs from 24 points of view
  Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy both conditions below:   (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal.   (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal and the matrix representing $A$ is irreducible tridiagonal.   We call such a pair a {\it Leonard pair} on $V$. Referring to the above Leonard pair, we investigate 24 bases for $V$ on which the action of $A$ and $A^*$ takes an attractive form. With respect to each of these bases, the matrices representing $A$ and $A^*$ are either diagonal, lower bidiagonal, upper bidiagonal, or tridiagonal. 
Paul Terwilliger