On divisible weighted Dynkin diagrams and reachable elements
Let e be a nilpotent element of a complex simple Lie algebra $ \mathfrak{g} $. The weighted Dynkin diagram of e, $ \mathcal{D}(e) $, is said to be divisible if $ {{{\mathcal{D}(e)}} \left/ {2} \right.} $ is again a weighted Dynkin diagram. The corresponding pair of nilpotent orbits is said to be friendly. In this paper we classify the friendly pairs and describe some of their properties. Any subalgebra $ \mathfrak{s}{\mathfrak{l}_3} $ in $ \mathfrak{g} $ gives rise to a friendly pair; such pairs are called A2-pairs. If Gx is the lower orbit in an A2-pair, then $ x \in \left[ {{\mathfrak{g}^x},{\mathfrak{g}^x}} \right] $, i.e., x is reachable. We also show that $ {\mathfrak{g}^x} $ has other interesting properties. Let $ {\mathfrak{g}^x} = { \oplus_{i \geqslant 0}}{\mathfrak{g}^x}(i) $ be the $ \mathbb{Z} - {\text{grading}} $ determined by a characteristic of x. We prove that $ {\mathfrak{g}^x} $ is generated by the Levi subalgebra $ {\mathfrak{g}^x}(0) $ and two elements of $ {\mathfrak{g}^x}(1) $. In particular, the nilpotent radical of $ {\mathfrak{g}^x} $ is generated by the subspace $ {\mathfrak{g}^x}(1) $.