The purpose of this work is to give some new algebraic properties of the orthogonality of a monic polynomial sequence {Qn } n≥0 defined by Qn (x) = Pn (x) + sn Pn−1 (x) + tn Pn−2 (x) + rn Pn−3 (x), n ≥ 1, where r n 0, n ≥ 3, and {Pn } n≥0 is a given sequence of monic orthogonal polynomials. Essentially, we consider some cases in which the parameters r n , s n , and t n can be computed more easily. Also, as a consequence, a matrix interpretation using LU and UL factorization is done. Some applications for Laguerre, Bessel and Tchebychev orthogonal polynomials of second kind are obtained.