a crossed product of a skew field of quaternions and four-group

The article considers construction of generalized crossed product of an arbitrary quaternions skew field and Klein four-group relative to factor system. It is well known that such crossed product is semisimple ring. Under specific conditions it is easy to show that crossed product of simple algebra and its inner automorphism group is central simple algebra. Finding out the conditions under which the crossed product is division algebra we face to difficulties of general case linked to analysis of system of linear equations defined over non-commutative rings. In the terms of anisotropic quadratic forms, there are sufficient conditions under those the crossed product of skew field of quaternions and four-group relative to factor system is division algebra. In addition, it is proved that the crossed product is the tensor product of two quaternions skew fields.