  Mizuno provided 19 examples of generalized rank three Nahm sums with
symmetrizer $\mathrm{diag}(1,1,2)$ which are conjecturally modular. We confirm
their modularity by establishing Rogers--Ramanujan type identities of index
$(1,1,2)$ for these examples. We first reduce these Nahm sums to some double
sums or single sums, and then we use known results or apply the theory of
Bailey pairs to prove the desired identities. Meanwhile, we generalize some
triple sum identities to general multi-sum identities.
