  Quaternions often appear in wide areas of applied science and engineering
such as wireless communications systems, mechanics, etc. It is known that are
two types of non-isomorphic generalized quaternion algebras, namely: the
algebra of quaternions and the algebra of coquaternions. In this paper, we
present the formulae to pass from a basis in the generalized quaternion
algebras to a basis in the division quaternions algebra or to a basis in the
coquaternions algebra and vice versa. The same result was obtained for the
generalized octonion algebra. Moreover, we emphasize the applications of these
results to the algebraic equations and De Moivre s formula in generalized
quaternion algebras and in generalized octonion division algebras.
