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Let ${\mathbf U}^-_q$ be the negative part of the quantized envelopingalgebra associated to a Kac-Moody algebra ${\mathfrak g}$ of symmetric type,and $\underline{\mathbf U}^-_q$ the algebra corresponding to the orbit algebra${\mathfrak g}^{\sigma}$ obtained from an admissible diagram automorphism$\sigma$ on ${\mathfrak g}$. Lusztig consructed the canonical basis ${\mathbfB}$ of ${\mathbf U}_q^-$ and the canonical signed basis$\underline{\widetilde{\mathbf B}}$ of $\underline{\mathbf U}_q^-$ by makinguse of the geometric theory of quivers. He proved that there is a naturalbijection $\widetilde{\mathbf B}^{\sigma} \to \widetilde{\underline{\mathbfB}}$. In this paper, assuming the existence of the canonical basis ${\mathbfB}$ of ${\mathbf U}_q^-$, we construct the canonical signed basis$\widetilde{\underline{\mathbf B}}$ of $\underline{\mathbf U}_q^-$, and anatural bijection $\widetilde{\mathbf B}^{\sigma} \to\widetilde{\underline{\mathbf B}}$ by an elementary method.

Diagram automorphisms and canonical bases for quantized enveloping algebras