Quasi-definite axial algebras of Jordan type half

Axial algebras are commutative nonassociative algebras generated by a finite set of primitive idempotents which action on an algebra is semisimple, and the fusion laws on the products between eigenvectors for these idempotents are fulfilled. We find the sufficient conditions in terms of the Frobenius form and of the properties of idempotents under which an axial algebra of Jordan type half is unital.Comment: 15 p.; v2: the second part of Section 8 is removed, since Lemma 21 in such form is not tru