Consider operators $A, B \, \& \, C$ which act on a vector space, and obey the following commutator relation.
$$[A, B] = 2C$$
Consider no restriction on their eigenvalue spectrum, i.e. the eigenvalues are unbounded. Given this 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?
$$[A', B'] = 2 C'$$
Is there a convenient algorithm to generate these new operator relations from the given commutator structure in the first equation?