We define the notion of a Lie superalgebra over a field k of characteristic 2 which unifies the two pre-existing ones – \(\mathbb {Z}/2\) -graded Lie algebras with a squaring map and Lie algebras in the Verlinde category \(\textrm{Ver}_4^+(k)\) , and prove the PBW theorem for this notion. We also do the same for the restricted version. Finally, we discuss mixed characteristic deformation theory of such Lie superalgebras (for perfect k), introducing and studying a natural lift of our notion of Lie superalgebra to characteristic zero – the notion of a mixed Lie superalgebra over a ramified quadratic extension R of the ring of Witt vectors W(k).
