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We present an elementary construction of a (highly degenerate) Hopf pairingbetween the universal enveloping algebra $U(\mathfrak{g})$ of afinite-dimensional Lie algebra $\mathfrak{g}$ over arbitrary field $\mathbf{k}$and the Hopf algebra $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$ of regularfunctions on the automorphism group of $\mathfrak{g}$. This pairing induces aHopf action of $\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$ on $U(\mathfrak{g})$which together with an explicitly given coaction makes $U(\mathfrak{g})$ into abraided commutative Yetter--Drinfeld$\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$-module algebra. From these data oneconstructs a Hopf algebroid structure on the smash product algebra$\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))\sharp U(\mathfrak{g})$ retainingessential features from earlier constructions of a Hopf algebroid structure oninfinite-dimensional versions of Heisenberg double of $U(\mathfrak{g})$,including a noncommutative phase space of Lie algebra type, while avoiding theneed of completed tensor products.  We prove a slightly more general result where algebra$\mathcal{O}(\mathrm{Aut}(\mathfrak{g}))$ is replaced by$\mathcal{O}(\mathrm{Aut}(\mathfrak{h}))$ and where $\mathfrak{h}$ is anyfinite-dimensional Leibniz algebra having $\mathfrak{g}$ as its maximal Liealgebra quotient.

Enveloping algebra is a Yetter--Drinfeld module algebra over Hopf algebra of regular functions on the automorphism group of a Lie algebra