Are all 12 possibilities for products of three octonions in fact possible?

Question

The octonions are a non-associative, non-commutative, $8$-dimensional algebra over the reals. Consider a triple $(x,y,z)$ of three distinct octonions. There are $12$ possible products of those, such as $(xy)z$, $x(yz)$, $(yx)z$, $y(xz)$, etc. Given such a triple, I define its "product number" to be the number of distinct possible products. So, to give an example, for a triple of real numbers, the product number is $1$. A priori, there are $12$ possibilities for product numbers, namely the positive integers from $1$ to $12$. Are all these $12$ possibilities in fact realized? Or, are some numbers "forbidden"? For each such number from $1$ to $12$ that is realized, I would like an explicit triple of three distinct octonions that realizes it. And, for each one that is never realized, I would like a proof that it is not realized. Of course, I could have chosen some arbitrary non-associative, non-commutative magma, but I am more interested in the case of the octonions.