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In this paper we determine the precise extent to which the classical    sl  2   ‐theory of complex semisimple finite‐dimensional Lie algebras due to Jacobson–Morozov and Kostant can be extended to positive characteristic. This builds on work of Pommerening and improves significantly upon previous attempts due to Springer–Steinberg and Carter/Spaltenstein. Our main advance arises by investigating quite fully the extent to which subalgebras of the Lie algebras of semisimple algebraic groups over algebraically closed fields   k   are   G  ‐completely reducible, a notion essentially due to Serre. For example, if   G   is exceptional andchark  =  p  ⩾  5   , we classify the triples    (  h  ,  g  ,  p  )    such that there exists a non‐  G  ‐completely reducible subalgebra of    g  =  Lie  (  G  )    isomorphic to   h  . We do this also under the restriction that   h   be a   p  ‐subalgebra of   g  . We find that the notion of subalgebras being   G  ‐completely reducible effectively characterises when it is possible to find bijections between the conjugacy classes of    sl  2   ‐subalgebras and nilpotent orbits and it is this which allows us to prove our main theorems.  For absolute completeness, we also show that there is essentially only one occasion in which a nilpotent element cannot be extended to an    sl  2   ‐triple when    p  ⩾  3   : this happens for the exceptional orbit in    G  2    when    p  =  3   .

The Jacobson–Morozov theorem and complete reducibility of Lie subalgebras