The stabilizer of two-dimensional vector space of 27-dimensional module of type e 6 over a field of characteristic two

The purpose of this paper is to use the notion of M-sets (cocliques) introduced by the second author in [S. Aldhafeeri and M. Bani-Ata, On the construction of Lie-Algebras of type E6(K) for fields of characteristic two, Beitrag Zur Algebra und Geometry 58 (2017) 529-534.] and using Levi components and unipotent radical subgroups of E6(K) to give an elementary and self-contained construction of the stabilizer of two dimensional vector space of 27-dimensional module of type E6 over a field of characteristic two. This stabilizer is in fact the maximal parabolic subgroup P2 of E6 or a Borel subgroup. This construction is elementary on the account that we use not more than little naive linear algebra notions. For more information one can see [M. E. Aschbacher, The 27-dimensional module for E6, 1, Invent. Math. 89 (1987) 159-195; M. E. Aschbacher, The 27-dimensional module for E6, II, J. London Math. Soc. 37 (1988) 275-293; M. Bani-Ata, On Lie algebras of type F4 and D4 over finite fields of characteristic two, Preprint; B. Cooperstein, Subgroups of the group E6(q) which are generated by root-subgroups, J. Algebra 46 (1977) 355-388.].