Automorphisms of free braided nonassociative algebras of rank 2

We prove the elementary reducibility of any nonaffine automorphism of a free nonassociative algebra of rank two over an arbitrary field. Using this result establish a property of automorphisms of this algebra that will be needed in later. We then derive a necessary and sufficient condition for the isomorphism of two free braided nonassociative algebras of rank two over a field with diagonal braidings. We describe the automorphism groups of two generated free braided nonassociative algebras with involutive diagonal braidings over an arbitrary field of characteristic not equal to two. Depending on the form of the diagonal involutive braiding, five different automorphism groups of a two-generated free nonassociative algebra arise in this case: 1) the group of all automorphisms, 2) the group of all odd automorphisms, 3) the subgroup of the group of triangular automorphisms, 4) the toric automorphism group, 5) the semidirect product of the toric automorphism group with the subgroup generated by an automorphism that permutes two variables.