In a Jordan algebra elements $a$ and $b$ are said to operator-commute, whenever $a \circ (b \circ x) = b \circ (a \circ x)$ for every other element $x$. (That is: $T_aT_b = T_bT_a$, writing $T_x(y) = x \circ y$.) In a JB-algebra elements $a$ and $b$ operator-commute if and only if they generate an associative subalgebra. (See e.g. p.44 "Jordan operator algebras" by Hanche-Olsen and Størmer.) Does this generalise from pairs to arbitrary subsets?
Question: Assume $A$ is a JB-algebra. $S \subseteq A$ is a subset of pairwise operator-commuting elements. Is the algebra generated by $S$ in $A$ associative?