Majorana algebras generated by a 2 ⁢ A {2A} algebra and one further axis

We consider Majorana algebras generated by three Majorana axes {a_{0}} , {a_{1}} and {a_{2}} such that {a_{0}} and {a_{1}} generate a dihedral algebra of type {2A} . We show that such an algebra must occur as a Majorana representation of one of 27 groups. These 27 groups coincide with the subgroups of the Monster that are generated by three {2A} -involutions a, b and c such that ab is also a {2A} -involution, which were classified by S. P. Norton in 1985. Our work relies on that of S. Decelle and consists of showing that certain groups do not admit Majorana representations.