  Nonassociative commutative algebras $A$ generated by idempotents $e$ whose
adjoint operators ${\rm ad}_e\colon A \rightarrow A$, given by $x \mapsto xe$,
are diagonalizable and have few eigenvalues are of recent interest. When
certain fusion (multiplication) rules between the associated eigenspaces are
imposed, the structure of these algebras remains rich yet rather rigid. For
example vertex operator algebras give rise to such algebras. The connection
between the Monster algebra and Monster group extends to many axial algebras
which then have interesting groups of automorphisms.
  Axial algebras of Jordan type $\eta$ are commutative algebras generated by
idempotents whose adjoint operators have a minimal polynomial dividing
$(x-1)x(x-\eta)$, where $\eta \notin \{0,1\}$ is fixed, with well-defined and
restrictive fusion rules. The case of $\eta \neq \frac{1}{2}$ was thoroughly
analyzed by Hall, Rehren, and Shpectorov in a recent paper, in which axial
algebras were introduced. Here we focus on the case where $\eta=\frac{1}{2}$,
which is much less understood and is of a different nature.
