  We initiate the investigation of the projective varieties $\mathbb
E(r,\mathfrak g)$ of elementary subalgebras of dimension $r$ of a
($p$-restricted) Lie algebra $\mathfrak g$ for various $r \geq 1$. These
varieties $\mathbb E(r,\mathfrak g)$ are the natural ambient varieties for
generalized support varieties for restricted representations of $\mathfrak g$.
We identify these varieties in special cases, revealing their interesting and
varied geometric structures. We also introduce invariants for a finite
dimensional $\mathfrak u(\mathfrak g)$-module $M$, the local $(r,j)$-radical
rank and local $(r,j)$-socle rank, functions which are lower/upper
semicontinuous on $\mathbb E(r,\mathfrak g)$. Examples are given of $\mathfrak
u(\mathfrak g)$-modules for which some of these rank functions are constant.
