We present an elementary construction of a (highly degenerate) Hopf pairing between the universal enveloping algebra $U(\mathfrak{ g})$ of a finite-dimensional Lie algebra $\mathfrak{ g}$ over arbitrary field $\mathbf{ k}$ and the Hopf algebra $\mathcal{ O}(\mathrm{ Aut}(\mathfrak{ g}))$ of regular functions on the automorphism group of $\mathfrak{ g}$. This pairing induces a Hopf action of $\mathcal{ O}(\mathrm{ Aut}(\mathfrak{ g}))$ on $U(\mathfrak{ g})$ which together with an explicitly given coaction makes $U(\mathfrak{ g})$ into a braided commutative Yetter--Drinfeld $\mathcal{ O}(\mathrm{ Aut}(\mathfrak{ g}))$-module algebra. From these data one constructs a Hopf algebroid structure on the smash product algebra $\mathcal{ O}(\mathrm{ Aut}(\mathfrak{ g}))\sharp U(\mathfrak{ g})$ retaining essential features from earlier constructions of a Hopf algebroid structure on infinite-dimensional versions of Heisenberg double of $U(\mathfrak{ g})$, including a noncommutative phase space of Lie algebra type, while avoiding the need 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 any finite-dimensional Leibniz algebra having $\mathfrak{ g}$ as its maximal Lie algebra quotient.
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