Normal elements of noncommutative Iwasawa algebras over SL₃(ℤ_p)

Let p be a prime integer and let ℤ_p be the ring of p-adic integers. By a purely computational approach we prove that each nonzero normal element of a noncommutative Iwasawa algebra over the special linear group SL₃ (ℤ_p) is a unit. This gives a positive answer to an open question in [F. Wei and D. Bian, Erratum: Normal elements of completed group algebras over SL_n (ℤ_p) [mr2747414], Internat. J. Algebra Comput. 23 (2013), no. 1, 215] and makes up for an earlier mistake in [F. Wei and D. Bian, Normal elements of completed group algebras over SL_n (ℤ_p), Internat. J. Algebra Comput. 20 (2010), no. 8, 1021-1039] simultaneously.