The adjoint representation matrix for the Lie pseudoalgebra (Q1358546)

The primary objects of the theory of pseudogroups in the sense of E. Cartan are left-invariant differential forms (analogous to Maurer-Cartan form of Lie groups) satisfying the structural equations \[ d\omega^\alpha =\Sigma C^\alpha_{\beta \gamma} \omega^\beta \wedge \omega^\gamma +\Sigma A^\alpha_{\beta a} \omega^\beta \wedge \omega^a\;(1\leq \alpha, \beta, \gamma< n_0,\;n_0\leq a<n_1) \] and their prolongations defined in an infinite-dimensional underlying space. Then the dual left-invariant vector fields constitute an infinite-dimensional Lie algebra with natural adjoint representation. The author explains these well-known results in terms of coordinates.