Hadamard algebras possessing the unique noncommutative simple component

The famous conjecture on the orders of Hadamard matrices may be reformulated as follows: a commutative algebra is Hadamard if and only if its dimension is divisible by $4$. This paper investigates the Hadamard algebras closed to commutative ones, namely, the algebras possessing the unique noncommutative simple component — the matrix algebra of order $2$.