  With each Steiner triple system there is associated a one-parameter family of
commutative, nonassociative, nonunital algebras that are by construction exact,
meaning that the trace of every multiplication operator vanishes, and these
algebras are shown to be Killing metrized, meaning the Killing type trace-form
is nondegenerate and invariant (Frobenius), and simple, except for certain
parameter values. The definition of these algebras resembles that of the Matsuo
algebra of the Steiner triple system, but they are different. For a Hall triple
system, the associated algebra is a primitive axial algebra for a
$\mathbb{Z}/2\mathbb{Z}$-graded fusion law.
