The structure of a semisimple Lie algebra $G$ can be described in terms of
its root system which is a finite set $\Sigma$ in a Euclidean space. These
systems play a fundamental role in the classical Killing-Cartan theory. The
structure of $G$ can be also characterized by a linearly independent subsystem
$\Pi$ of $\Sigma$ -- a simple root system. In 1946 Dynkin introduced graphs
describing simple root systems. They are widely used under the name Dynkin
diagrams. In this paper we introduce an enhancement of these diagrams.
Ingredients of enhanced graphs are the same as in Dynkin diagrams: some of
nodes are connected by single, double or triple bonds. However nodes represent
not single roots $\alpha$ but pairs $(\alpha,-\alpha)$. We call such pairs
projective roots. We use enhanced diagrams to classify Weyl orbits in the space
of subsets of $\Sigma$. A special role is played by subsets $\Lambda$ with the
property: $\alpha-\beta\notin\Sigma$ if $\alpha,\beta\in\Lambda$. The Weyl
group $W$ of $G$ preserves the space $\mathbb{P}$ of such sets, and its orbits
in $\mathbb{P}$ are in 1-1 correspondence with classes of conjugate regular
semisimple subalgebras of $G$. As another new tool we introduce maximal
orthogonal subsets $M$ of $\Sigma$. We call the subgroup of the Weyl group
preserving $M$ a core group of $G$. We represent the core group of $E_7$ as the
group of linear transformations and the core group of $E_8$ as the group of
affine transformations in the 3-dimensional linear space over the field
$\mathbb{Z}_2$. We use these representations and appropriate representations
for the core groups of $A_n$, $D_n$, $E_6$ in our classification of Weyl
orbits. For algebras $B_n$, $C_n$, $G_2$, $F_4$ additional considerations are
needed. They will be the subject of Part II.