In this paper we show that strong noncommutative Jordan algebras R over an arbitrary ring of scalars having the alternator mappings y,y,-1 as Jordon derivations are U-algebras, algebras such that Uablpar;crpar; lies in the Jordan ideal generated by a. For any U-algebra R we relate the radical theories of R and R+. Our main result is that any radical property p′ of U-algebras such that P′-radR⊂ p-radR+. If p is nondegenerate the P′ is nondegenerate and P′-radR=p-radR+. This applies in particular to the McCrimmon, locally nilpotent, nil, Jacobson and Brown-McCoy radicals of Jordan algebras