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



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> What is the minimum number of generators of this Jordan algebra? And could you give me one set of such generators?
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Thanks a lot!



Generators for the Jordan algebra of symmetric 3-by-3 matrices
The minimum number of generators is 2. First, it is easy to check that one generator is not enough: every symmetric matrix is diagonalizable, so the subalgebra it generates has dimension at most 3. 


Next, the claim is that $$A:=S\_{11}+2S\_{22}+3S\_{33}$$ and $$B:=S\_{12}+2S\_{13}$$ generate the algebra, where $S\_{ij}$ is the usual notation for the symmetric matrices. In fact the powers of $A$ generate the subalgebra $D$ of diagonal matrices (by Vandermonde) and the powers $B$ satisfy $B^2=S\_{23}$ mod $D$ and $B^3=S\_{12}+\frac12S\_{13}$ mod $D$.


