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Axial algebras are a class of commutative non-associative algebras generatedby idempotents, called axes, with adjoint action semi-simple and satisfying aprescribed fusion law. Axial algebras were introduced by Hall, Rehren andShpectorov \cite{hrs,hrs1} as a broad generalisation of Majorana algebras ofIvanov, whose axioms were derived from the properties of the Griess algebra forthe Monster sporadic simple group. The class of axial algebras of Monster typeincludes Majorana algebras for the Monster and many other sporadic simplegroups, Jordan algebras for classical and some exceptional simple groups, andMatsuo algebras corresponding to $3$-transposition groups. Thus, axial algebrasof Monster type unify several strands in the theory of finite simple groups.  It is shown here that double axes, i.e., sums of two orthogonal axes in aMatsuo algebra, satisfy the fusion law of Monster type $(2\eta,\eta)$.Primitive subalgebras generated by two single or double axes are completelyclassified and $3$-generated primitive subalgebras are classified in one of thethree cases. These classifications further lead to the general flipconstruction outputting a rich variety of axial algebras of Monster type. Anapplication of the flip construction to the case of Matsuo algebras related tothe symmetric groups results in three new explicit infinite series of suchalgebras.

Double axes and subalgebras of Monster type in Matsuo algebras