  In this paper we obtain the LU-decomposition of a noncommutative linear
system of equations that, in the rank one case, characterizes the image of the
Lepowsky homomorphism $U(\lieg)^{K}\to U(\liek)^{M}\otimes U(\liea)$. This
LU-decomposition can be transformed into very simple matrix identities, where
the entries of the matrices involved belong to a special class of Jacobi
polynomials. In particular, each entry of the L part of the original system is
expressed in terms of a single ultraspherical Jacobi polynomial. In turns,
these matrix identities yield a biorthogonality relation between the
ultraspherical Jacobi polynomials.
