CHSH test using unbounded operators

I want to perform a CHSH inequalities test using operators $A , & B$ and their combinations each possessed by Alice and Bob, which obey the following commutator relation.

$$[A, B] = 2C$$

Consider that the eigenvalue spectrum of all three operators is unbounded. I now want to recast them in the standard CHSH form by defining new operators such that their norm is less than or equal to unity. Given that the above operator relation holds, do there exist operators $A’$, $B’$ and $C’$ such that their eigenvalues lie between $-1$ and $1$, i.e.

$$||A’||leq 1,quad ||B’||leq 1, quad ||C’|| leq 1,$$

such that
$$A’ = f_1(A), quad B’ = f_2(B), quad C’ = f_3(C)$$

and the following relation holds?

$$[A’, B’] = 2 C’$$

Note that this transformation will allow me to write down operators which replicate standard spin operators, using which the CHSH bound of "2" can be possibly violated on an entangled state in textbook fashion, as this relation is now similar to relation between spin generators. Is there a convenient algorithm to generate these new operator relations from the given commutator structure in the first equation? Or is there an easier way to perform the CHSH test using operators in the first equation?