Quotient algebra of tensor algebra, waiting with the quotients?

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I was just wondering if we in a quotient algebra of a tensor algebra can "wait" with the quotients until we have computed the expression in the tensor algebra? This would make it simpler for my intuition if we when doing nested expressions can pretend we live in tensor algebra land and then in the end search through all the terms and see if they can be reduced with the quotient algebra rules. Can we even multiply a quotient algebra element with a normal vanilla tensor algebra element?