The author considers four examples of partial differential equations with constant coefficients, including the so-called \(4\)-dimensional universal hierarchy \[ u_{zz} = u_{tx} +u_z u_{xy} - u_x u_{yz}, \] where the subscript denotes partial derivatives in the corresponding variables. Here \(u\) is a real analytic function on \(\mathbb{R}^4\). The considered equations are known to admit Lax representations with a non-removable spectral parameter \(\lambda\). In the case of the above equation, the Lax representation is given by \[ \begin{cases} v_t =\lambda^2 v_x - (\lambda u_x +u_z) v_y,\\ v_z=\lambda v_x- u_x v_y . \end{cases} \] In the four cases under study, the author shows that the Lax representations can be deduced by algebraic methods. More precisely, each equation determines an infinite-dimensional Lie algebra, the so-called contact symmetry algebra, containing as a semi-direct summand a finite-dimensional non-abelian Lie algebra. The exotic cohomology in dimension two of this algebra turns out to be non-trivial, producing non-central extensions of the contact symmetry algebra. Finally, the Lax representation arises as the Maurer-Cartan form of these extensions.