Let F denote a field, and let V denote a vector space of finite positive dimension over F. Let A, A * denote a tridiagonal pair on V of diameter d 3. Assume the eigenvalue and dual eigenvalue sequences of A, A * satisfy θ i = q 2i θ , θ * i = q 2d−2i θ * for some nonzero scalars θ , θ * , q ∈ F, where q is not a root of unity. Assume that not all eigenvalues of A and A * have multiplicity one. Let M and M * denote the subalgebras of End(V ) generated by A and A * , respectively, and assume that V = Mv * + M * v for some eigenvectors v * of A * associated with θ * 0 and v of A associated with θ d . We find a nice basis for V and describe the action of A, A * on this basis in terms of six parameters.
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