Lie algebras with a subalgebra of codimension $p$

All Lie algebras over an algebraically closed field $\mathbf K$, with $\operatorname{char}\mathbf K=p>3$, which have a faithful irreducible representation of degree $p$ are enumerated. Graded Lie algebras $L=\bigoplus^r_{i=-q}L_i$, which have subalgebra $L^-=\bigoplus_{i<0}L_i$ with $\operatorname{dim}L^-=p$ are investigated. Simple finite-dimensional modular Lie algebras which have a maximal subalgebra $\mathscr L_0$ of codimension $p>5$ such that for the corresponding noncontractible filtration with $\mathscr L_1\ne0$ the algebra $\operatorname{Gr}\mathscr L$ is transitive are characterized as deformations of such graded algebras.
Bibliography: 15 titles.