Recently, Paul Terwilliger introduced the notion of a lowering–raising (or LR) triple, and classified the LR triples. An LR triple is defined as follows. Fix an integer d≥0, a field cd82611df19dacd343c839" title="Click to view the MathML source">F, and a vector space V over cd82611df19dacd343c839" title="Click to view the MathML source">F with dimension d+1. By a decomposition of V we mean a sequence cdc8c41125dde372b69141fea"> of 1-dimensional subspaces of V whose sum is V. For a linear transformation A from V to V, we say A lowers cdc8c41125dde372b69141fea"> whenever AVi=Vi−1 for 0≤i≤d, where cde9fb3a196362" title="Click to view the MathML source">V−1=0. We say A raises cdc8c41125dde372b69141fea"> whenever AVi=Vi+1 for 0≤i≤d, where Vd+1=0. An ordered pair of linear transformations A, B from V to V is called LR whenever there exists a decomposition cdc8c41125dde372b69141fea"> of V that is lowered by A and raised by B . In this case the decomposition cdc8c41125dde372b69141fea"> is uniquely determined by A, B ; we call it the (A,B)-decomposition of V. Consider a 3-tuple of linear transformations A, B, C from V to V such that any two of A, B, C form an LR pair on V. Such a 3-tuple is called an LR triple on V. Let 伪, 尾, 纬 be nonzero scalars in cd82611df19dacd343c839" title="Click to view the MathML source">F. The triple 伪A, 尾B, 纬C is an LR triple on V, said to be associated to A, B, C . Let cdc8c41125dde372b69141fea"> be a decomposition of V and let X be a linear transformation from V to V. We say X is tridiagonal with respect to cdc8c41125dde372b69141fea"> whenever XVi⊆Vi−1+Vi+Vi+1 for 0≤i≤d. Let X be the vector space over cd82611df19dacd343c839" title="Click to view the MathML source">F consisting of the linear transformations from V to V that are tridiagonal with respect to the (A,B) and (B,C) and (C,A) decompositions of V. There is a special class of LR triples, called q -Weyl type. In the present paper, we find a basis of X for each LR triple that is not associated to an LR triple of q-Weyl type.