Let $$F = \mathbb{Q} (\sqrt{-d_1})$$ and $$E = \mathbb{Q} (\sqrt {-d_1}, \sqrt{d_2})$$ , d 1 and d 2 squarefree integers, be an imaginary field and a biquadratic field, respectively. Let S be the set consisting of all infinite primes, all dyadic primes and all finite primes which ramify in E. Suppose the 4-rank of the class group of F is zero and the S-ideal class group of F has odd order, we give the forms of all elements of order ⩽ 2 in K 2 O E and use the Hurrelbrink and Kolster’s method [Hurrelbrink, J. and Kolster, M.: J. reine angew. Math. 499 (1998), 145–188] to obtain the forms of all elements of order 4 in K 2 O E .