The reduced Gassner representation is a multi-parameter representation of P n , the pure braid group on n strings. Specializing the parameters t 1 , t 2 , . . . , t n to nonzero complex numbers z 1 , z 2 , . . . , z n gives a representation G n (z 1 , . . . , z n ) : It was proved that the representation is irreducible if and only if z 1 . . . z n = 1. In our work, we consider the case n = 4 and we determine the composition factors of G 4 (z 1 , . . . , z 4 ) when it is reducible. Our main

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... em shows that the reduced Gassner representation G 4 (z 1 , . . . , z 4 ) of degree three is either a direct sum of one-dimensional representations or it has a composition factor of degree 2, namely, G 4 (z 1 , . . . , z 4 ) , which is an extension of the irreducible representation G 3 (z 1 , z 2 , z 3 ) to P 4 .