  Axial algebras are non-associative algebras generated by semisimple
idempotents, known as axes, that all obey a fusion rule. Axial algebras were
introduced by Hall, Rehren and Shpectorov as a generalisation of the axioms of
Majorana theory, which was in turn introduced by Ivanov as an axiomatisation of
certain properties of the 2A-axes of the Griess algebra. Axial algebras of
Monster type are axial algebras whose axes obey the Monster, or Majorana,
fusion rule.
  We construct an axial algebra of Monster type $M_{4A}$ over the polynomial
ring $\mathbb{R}[t]$ that is generated by six axes whose Miyamoto involutions
generate an elementary abelian group of order $4$. This construction
automatically provides an infinite-parameter family $\{M(t)\}_{t \in
\mathbb{R}}$ of axial algebras of Monster type each of which admit a unique
Frobenius form. Moreover, we show that this form on $M(t)$ is positive definite
if and only if $0 < t < \frac{1}{6}$ and also satisfies Norton's inequality if
and only if $0 \leq t \leq \frac{1}{6}$. Finally, we show that the $4A$ axes of
$M_{4A}$ obey a $C_2 \times C_2$-graded fusion rule giving a new infinite
family of fusion rules.
