Diagram automorphisms and canonical bases for quantum affine algebras, II

Let ${\mathbf{U}}_q^{-}$ be the negative part of the quantum enveloping algebra, and $\sigma$ the algebra automorphism on ${\mathbf{U}}_q^{-}$ induced from a diagram automorphism. Let ${\underset{‾}{\mathbf{U}}}_q^{-}$ be the quantum algebra obtained from $\sigma$, and $\tilde{\mathbf{B}}$ (resp. $\widetilde{\underset{‾}{\mathbf{B}}}$) the canonical signed basis of ${\mathbf U}_q^-$(resp\mathrm{.}$\underline{\mathbf U}_q^-$)\mathrm{.}Assumethat${\mathbf U}_q^-$ is simply-laced of finite or affine type. In our previous papers [SZ1, 2], we have proved by an elementary method, that there exists a natural bijection ${\tilde{\mathbf{B}}}^{\sigma }\simeq \widetilde{\underset{‾}{\mathbf{B}}}$ in the case where $\sigma$ is admissible. In this paper, we show that such a bijection exists even if $\sigma$ is not admissible, possibly except some small rank cases.