A Clifford algebra associated to generalized Fibonacci quaternions

In this paper, using the construction of Clifford algebras, we associate to the set of generalized Fibonacci quaternions a quaternion algebra A (i.e., a Clifford algebra of dimension four). Indeed, for the generalized quaternion algebra H(beta(1), beta(2)), denoting E(beta(1), beta(2)) = 1/5[1 + beta(1) + 2 beta(2) + 5 beta(1)beta(2) + alpha(beta(1) + 3 beta(2) + 8 beta(1)beta(2))], if E(beta(1), beta(2)) > 0, therefore the algebra A is split. If E(beta(1), beta(2)) < 0, then the algebra A is a division algebra. In this way, we provide a nice algorithm to obtain a division quaternion algebra starting from a quaternion non-division algebra and vice versa.