gr.group theory - Finite index normal subgroups of a free group.

Let $G= (mathbb{Z} bigoplus mathbb{Z}) star (mathbb{Z} bigoplus mathbb{Z})$, where $star$ denotes the free product, let F be the commutator subgroup of G, it is free by a theorem of Kurosh. Find a proper normal subgroup of F (other than the trivial one) such that it is of infinite index.

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