Gelu Popescu has a big paper on multivariable generalizations of the numerical radius (Memoirs of the AMS, arXiv). In page 8 of this memoirs I see the following paragraph:
Are the operators $S_i$ commuting?
2026-03-29 13:46:45.1774792005
The left creation operators on the Fock space
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Well, $S_i(S_j1)=S_i(e_j)=e_i\otimes e_j$ and $S_j(S_i1)=e_j\otimes e_i$. In $H\otimes H$, $e_i\otimes e_j\ne e_j \otimes e_i$ unless $i= j$, so if $i\ne j$ then $S_iS_j\ne S_j S_i$.