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Root vectors are important to understand quantized enveloping algebras. In this paper we establish a commutation formula for root vectors. By means of the formula we show that particular orders on root system are not necessary in constructing some integral bases of a quantized enveloping algebra (Theorem 2.4). Moreover using the formula we can show that certain PBW bases are orthogonal bases of the bilinear form considered by Kashiwara in his work on crystal bases, see 3.9.

A commutation formula for root vectors in quantized enveloping algebras