Say I have a finite collection $\{O_i\}_{i=1}^{d}$ of traceless, mutually orthogonal, dichotomic (meaning eigenvalues are $\pm 1$) observables satisfying the anti-commutation relations $\{O_i,O_j\}=2\delta_{ij}I$, where $I$ is the identity matrix. Assuming $d>2$, are these enough conditions to conclude that the algebra spanned by $\{O_i\}_{i=1}^{d}$ is isomorphic to the algebra spanned by the Pauli spin matrices? If not, what further conditions are needed to draw such a conclusion?
By the way, this is not a homework problem. I have come across such a collection of observables in my research, and am hoping to conclude that the algebra spanned by these observables is necessarily isomorphic to the algebra spanned by the Pauli spin matrices.
