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Let $Gamma$ be the first Grigorchuk group‎. ‎According to a result of Bar-thol-di‎, ‎the only left Engel elements of $Gamma$ are the involutions‎. ‎This implies that the set of left Engel elements of $Gamma$ is not a subgroup‎. ‎The natural question arises whether this is also the case for the sets of bounded left Engel elements‎, ‎right Engel elements and bounded right Engel elements of $Gamma$‎. ‎Motivated by this‎, ‎we prove that these three subsets of $Gamma$ coincide with the identity subgroup‎.