TITLE: Generators for the Jordan algebra of symmetric 3-by-3 matrices
QUESTION [6 upvotes]: In the vector space $V$ of $3\times 3$ symmetric real matrices, we can define a nonassociative algebra structure by the multiplication
$$A \bullet B = \frac12 (AB +BA).$$
This turns $V$ into a Jordan algebra.
Question

What is the minimum number of generators of this Jordan algebra?  And could you give me one set of such generators?

Thanks a lot!

REPLY [2 votes]: This is already well answered, many choices of two symmetric matrices will do. I'll add this observation: $\{I,A,B,A \bullet A,B \bullet B,A\bullet B\}$ is an additive basis where
$$A=\left[\begin{array}{ccc} 1 & 1 &1 \\ 1 & 0 & 0 \\ 1 & 0 & 0\end{array}\right] \qquad B=\left[\begin{array}{ccc} 0 & 0 &1 \\ 0 & 0 & 1 \\ 1 & 1 & 1\end{array}\right]$$ Note that $A\bullet B$ has rank $3$, not that it matters. I'm not sure if two rank 1 matrices could generate the algebra, but I lean against (I couldn't get past dimension 4), I didn't look hard for a generating set with a rank 1 and a rank 2 matrix.