Abstract. It is shown that in the algebra freely generated by $a$, $b$ and $c$ satisfying the commutation relations $[a,b]=ab-ba=c$, $[a,c]=[b,c]=0$ the identity $(abc)^n=a^nb^nc^n$ holds for every $n=0,1,\ldots $. Some applications of this assertion are also pointed out.

AMS Subject Classification (1991): 17B01, 17B35, 33C45, 34A05, 81S05

Received December 30, 1993. (Registered under 5608/2009.)