Majorana algebras generated by 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 which 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.