  Axial algebras are a class of commutative algebras generated by idempotents,
with adjoint action semisimple and satisfying a prescribed fusion law. Axial
algebras were introduced by Hall, Rehren, and Shpectorov in 2015 as a broad
generalization of Majorana algebras of Ivanov, whose axioms were derived from
the properties of the Griess algebra for the Monster group.
  The class of Matsuo algebras was introduced by Matsuo and later generalized
by Hall, Rehren, and Shpectorov. A Matsuo algebra $M$ is built by a set of
3-transpositions $D$. Elements of $D$ are idempotents in $M$ and called axes.
In particular, $M$ is an example of an axial algebra. It is known that double
axes, i.e., sums of two orthogonal axes in a Matsuo algebra, satisfy the fusion
law of Monster type. This observation shows that a set consisting of axes and
double axes can generate a subalgebra of Monster type in the Matsuo algebra.
Subalgebras corresponding to various series of 3-transposition groups are
extensively studied by many authors.
  In this paper, we study primitive subalgebras generated by a single axis and
two double axes. We classify all such subalgebras in seven out of nine possible
cases for a diagram on 3-transpositions that are involved in the generating
elements. We also construct several infinite series of axial algebras of
Monster type generalizing our 3-generated algebras.
