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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 $U(g)^{K} o U(k)^{M}otimes U(a)$.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.