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Let $G$ be the adjoint group of a real simple Lie algebra $\mathfrak{g}_0$equal either $\mathfrak{s}\mathfrak{u}(n,1)$ or$\mathfrak{s}\mathfrak{o}(n,1),$ $K$ its maximal compact subgroup, ${\calU}(\mathfrak{g})$ the universal enveloping algebra of the complexification$\mathfrak{g}$ of $\mathfrak{g}_0$ and ${\cal U}(\mathfrak{g})^K$ itssubalgebra of $K-$invariant elements. By a result of F. Knopp [3] ${\calU}(\mathfrak{g})$ is free as a ${\cal U}(\mathfrak{g})^K-$module, so thereexists a $K-$submodule $E$ of ${\cal U}(\mathfrak{g})$ such that themultiplication defines an isomorphism of $K-$modules ${\calU}(\mathfrak{g})^K\otimes E\longrightarrow{\cal U}(\mathfrak{g}).$ We provethat $E$ is equivalent to the regular representation of $K,$ i.e. that themultiplicity of every $\delta\in\hat{K}$ in $E$ equals its dimension. As aconsequence we get that for any finitedimensional complex $K-$module $V$ thespace $({\cal U}(\mathfrak{g})\otimes V)^K$ of $K-$invariants is free ${\calU}(\mathfrak{g})^K-$module of rank $\dim\,V.$

$K-$structure of ${\cal U}(\mathfrak{g})$ for $\mathfrak{s}\mathfrak{u}(n,1)$ and $\mathfrak{s}\mathfrak{o}(n,1)$