In the vector space \(\mathcal{P}\) of polynomials with coefficients in \(\mathbb{C}\), a sequence \(\{P_n\}_{n\geq 0}\) is called a polynomial sequence if \(\text{deg} \,P_n=n\), for any \(n\geq 0\). In the dual space \(\mathcal{P}'\) formed by all the linear functionals \(u:\mathcal{P}\longrightarrow \mathbb{C}:p\mapsto\langle u,p\rangle \), there exists for each monic polynomial sequence \(\{P_n\}_{n\geq 0}\) a unique dual sequence \(\{u_n\}_{n\geq 0}\) such that \(\langle u_n,P_m\rangle=\delta_{n,m}\). For each linear operator \(T:\mathcal{P}\longrightarrow \mathcal{P}\), the relation \(\langle {}^tT(u),p\rangle=\langle u,T(p)\rangle \) defines a linear operator \({}^tT:\mathcal{P}'\longrightarrow \mathcal{P}'\) called the transpose of \(T\). The differential operator \(D:\mathcal{P}\longrightarrow \mathcal{P}:p\mapsto D(p)\), where \(D(p)(x)=p'(x)\), plays a fundamental role. We say that the orthogonal polynomial sequence \(\{P_n\}_{n\geq 0}\) is a classical sequence in the Hahn’s sense if \(\{ (n\!+\!1)^{-1}DP_{n+1} \}_{n\geq 0}\) is also orthogonal. In this paper, the authors investigate the two-orthogonal and \(\Lambda \)-Appell monic sequences \(\{P_n\}_{n\geq 0}\) in the case when \(\Lambda \) is the lowering differential operator \(\Lambda =a_0D-3a_2DxD+a_2(Dx)^2D\). They obtain a matrix differential identity satisfied by the dual sequence, proving that the resultant polynomial is a classical sequence in the Hahn’s sense.