M_3(O) matrix algebra

$M_{3}(\mathbb O)$ consists of $3$ -by- $3$ matrices with entries in O, with entrywise addition and scalar multiplication and product

(XY)_{ij}=\sum_{r=1}^{3}X_{ir}Y_{rj}.

Its unit is $I_{3}$ . The scalar field for this catalogue object is $\mathbb R$ .

Dimension and multiplication

The real dimension is $72$ , since there are $9$ entries with $8$ real coordinates each.

Two-factor matrix products are unambiguous, but multiplication is not associative: the coefficient algebra embeds in a diagonal corner, so its nonzero associators survive. No associative-algebra or ring view is assigned. Symmetrizing this product must not be assumed to give a Jordan algebra.

Size convention

For every size at least two, $E_{11}E_{22}=0$ exhibits nonzero zero divisors. Matrix size is not the dimension of the whole algebra.