  Axial algebras are a recently introduced class of non-associative algebra,
with a naturally associated group, which generalise the Griess algebra and some
key features of the moonshine VOA. Sakuma's Theorem classifies the eight
$2$-generated axial algebras of Monster type. In this paper, we compute almost
all the $3$-generated axial algebras whose associated Miyamoto group is minimal
$3$-generated (this includes the minimal $3$-generated algebras). We note that
this work was carried out independently to that of Mamontov, Staroletov and
Whybrow and extends their result by computing more algebras and not assuming
primitivity, or an associating bilinear form.
