On centerless unimodular contact Lie algebras

We provide an elementary proof that, in a (transversely) unimodular contact Lie algebra, the adjoint action of the Reeb vector is nilpotent except when the Lie algebra is isomorphic to either $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$ or $\mathfrak{s}\mathfrak{u}(2)$. We introduce a class of contact Lie algebras, called $\mathit{\text{DS-contact}}\mathit{\text{Lie}}\mathit{\text{algebras}}$, containing all K-contact Lie algebras, and deduce from the previous result that the only centerless unimodular examples in this class are precisely $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$ and $\mathfrak{s}\mathfrak{u}(2)$. This gives an alternative proof of the previously known fact that centerless unimodular Sasakian Lie algebras are isomorphic to either $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$ or $\mathfrak{s}\mathfrak{u}(2)$. Some other results known to hold for Sasakian Lie algebras are generalized as well. We investigate several properties of DS-contact Lie algebras in relation to Frobenius Lie algebras, and also classify them in dimension five. Some implications for the contact Lefschetz condition are explored.