Okubo algebra

Remarks

The product stays in $V$: the adjoint of $\mu xy + \bar{\mu} yx$ is $\bar{\mu} yx + \mu xy$, and the trace of $x * y$ is $(\mu + \bar{\mu} - 1)\tr(xy) = 0$. The algebra is flexible but not unital, not associative and not alternative. There is a norm $N$ on $V$, a positive multiple of $x \mapsto \tr(x^2)$, with $N(x * y) = N(x)N(y)$ and $(x * y) * x = x * (y * x) = N(x)\,y$. So it is a composition algebra without a unit. Since $N$ is positive definite, $x * y = 0$ forces $x = 0$ or $y = 0$, so the Okubo algebra is an $8$-dimensional real division algebra that is not isomorphic to the octonions.