Bidiagonal triads and the tetrahedron algebra

We introduce a linear algebraic object called bidiagonal triad. A triad consists of three diagonalizable transformations on finite-dimensional vector space which satisfy the following condition. The eigenspaces can be ordered so that each transformation raises other two in block-bidiagonal fashion. show eigenvalue sequences associated to same recurrence relation. Based solutions this we define notion reduced triad, and every is equivalent one. irreducible representation tetrahedron Lie algebra provides numerous examples triads. Conversely, how representations constructed starting from certain type