The hyperquaternion algebra being defined as a tensor product of quaternion algebras (or a subalgebra thereof), it follows that Clifford algebras are hyperquater-nion algebras due to a theorem by Clifford (1878). Examples of hyperquaternionsare quaternions H(e1=i; e2=j); biquaternions HC(e1=iI; e2=jI; e3=kI); tetraquaternions HH(e0=j; e1=kI;e2=kJ; e3=kK); and so on HHC; HHH:::. The formula of n dimensional rotations in euclidean spaces y=axa-1 (a2C+n) was given by Lipschitz (1880). Moore, working on a canonical decomposition of rotations was to call the elements of Lipschitz’s algebra hyperquaternions whichjustifies the above terminology. Hyperquaternions provide a new efficient mathematical formalism for physics. Since HH≃M4 (R), [HH] C≃M4(C); [HH] H≃M4(H) it follows that hyperquaternions yield all real matrices as well as the complex and quaternionic ones. A hyperconjugation defined as A h=HcHcHc:::Hc where c indicates a quaternion conjugation, yields respectively the matrix transposition HcHc≃[M4(R)]t; the adjunction HcHcCc≃[M4(C)]y, and the transpose quaternion conjugate HcHcHc≃[M4(H)]tc: Using standard definitions, one then obtains simple hyperquaternion expressions of major symmetry groups of physics such as the orthogonal groups, the unitary and unitary symplectic groups as well as the extended Poincaré group (via dual hyperquaternions) and the conformal group.References:[1] Girard, Patrick R. et al., Differential Geometry Revisited by Biquaternion Clifford Algebra. In J.-D. Boissonat et al (Eds.): Curves and Surfaces (Springer, 2015).[2] Girard, Patrick R., Quaternions, Clifford Algebras and Relativistic Physics (Birkhauser, Basel, 2007).[3] Clifford, W. K., Applications of Grassmann’s extensive algebra, Amer. J. Math., 1 (1878), pp. 350-358.[4] Lipschitz R., Principes d’un calcul algébrique qui contient comme espèces particulières le calcul des quantités imaginaires et des quaternions, C. R. Acad. Sci. Paris, 91 (1880), pp. 619-621, 660-664.[5] Moore, C. J. E., Hyperquaternions, Journal of Mathematical Physics, 1 (1922),pp. 63-77.[6] Hankins, Thomas L., Sir William Rowan Hamilton (The Johns Hopkins University Press, Baltimore, 1980).