Hello!

I stumbled upon the following commutative, non-associative, three-dimensional algebra (with basis $\{A, B, C\}$):

$A\times A = 0$

$A\times B = A$

$A\times C = 2B$

$B\times A = A$

$B\times B = B$

$B\times C = C$

$C\times A = 2B$

$C\times B = C$

$C\times C = 0$

Is anything known about its irreducible representations? In particular, how many nonequivalent irreps do exist? What are their dimensions? Can we construct explicit representation matrices for a given irrep?