On numerical characteristics of subvarieties for three varieties of Lie algebras

Let $\mathbf V$ be a variety of Lie algebras. For each $n$ we consider the dimension of the space of multilinear elements in $n$ distinct letters of a free algebra of this variety. This gives rise to the codimension sequence $c_n(\mathbf V)$. To study the exponential growth one defines the exponent of the variety $\operatorname{Exp}\mathbf V=\varlimsup_{n\to\infty}\root n\of{c_n(\mathbf V)}$. The variety of Lie algebras with nilpotent derived subalgebra $\mathbf N_s\mathbf A$ is known to have $\operatorname{Exp}(\mathbf N_s\mathbf A)=s$. Over a field of characteristic zero the exponent of every subvariety $\mathbf V\subset \mathbf N_s\mathbf A$ is known to be an integer. We shall prove that this is true over any field.
Unlike associative algebras, for varieties of Lie algebras it is typical to have superexponential growth for the codimension sequence. Earlier the author introduced a scale for measuring this growth. The following extreme property is established for two varieties $\mathbf{AN}_2$ and $\mathbf A^3$. Any subvariety in each of them cannot be “just slightly smaller” in terms of this scale. That is, either a subvariety lies at the same point of the scale as the variety itself, or it is situated substantially lower on the scale. These results are also established over an arbitrary field and without using the representation theory of symmetric groups.