On the recognition theorem for Lie algebras of characteristic three

The finite-dimensional simple Lie algebras over an algebraically closed field of characteristic $p=3$ that admit a grading $(L_i;i\geqslant-1)$ of depth 1 are classified in this paper. It is assumed that $L_0$ is a reductive Lie algebra acting irreducibly on $L_{-1}$. Most of the arguments work for any characteristic $p\ne 2$. The case of a non-restricted $L_0$-module $L_{-1}$ was considered previously.