On root-involutions and root-subgroups of E ₆ (K) for fields K of characteristic two

The purpose of this paper is to investigate the root-involutions and root-subgroups of the Chevalley group E6(K) for fields K of characteristic two. The approach we follow is elementary and self-contained depends on the notion of M-sets which we have introduced in [Aldhafeeri and M. Bani-Ata, On the construction of Lie-algebras of type E6(K) for fields of characteristic two, Beit. Algebra Geom. 58 (2017) 529-534]. The approach is elementary on the account that it consists of little more than naive linear algebra. It is remarkable to mention that Chevalley groups over fields of characteristic two have not much been researched. This work may contribute in this regard. This paper is divided into three main sections: the first section is a combinatorial section, the second section is on relations among M-sets, the last one is on Lie algebra.