The resulting algebra is again a division algebra, the octonions O, constructed independently by Graves in 1843 and Cayley in 1845.
However, the 16-dimensional algebra obtained from O in this manner is no longer a division algebra.
Every associative finite-dimensional real division algebra is isomorphic to one of R, C, and H .
A new era in the theory of real division algebras was launched by Hopf  in 1940, when he proved that a finite-dimensional commutative real division algebra has either dimension one or two, and furthermore, that the dimension of any real division algebra is either a power of two or infinite.
Because the octavian algebra is a division algebra [1,2] then for each octavians u and [u.
But in accordance with the Hurwitz theorem * and with the generalized Frobenius theorem ([dagger]) a more than 8-dimensional Cayley-Dickson algebra does not a division algebra.
dagger]) division algebra can be only either 1 or 2 or 4 or 8-dimensional .
One potential difficulty with BQM is that the biquaternions are not a division algebra
It is known that there exist topological division algebras
which are not topologically isomorphic to C (see, for example, , pp.
Among the topics are the functoriality of Rieffel's generalized fixed-point algebras for proper actions, division algebras
and supersymmetry, Riemann-Roch and index formulae in twisted K-theory, noncommutative Yang-Mills theory for quantum Heisenberg manifolds, distances between matrix algebras that converge to coadjoint orbits, and geometric and topological structures related to M-branes.
This time it was decided to expand the scope by including some further topics related to interpolation, such as inequalities, invariant theory, symmetric spaces, operator algebras, multilinear algebra and division algebras, operator monotonicity and convexity, functional spaces and applications and connections of these topics to nonlinear partial differential equations, geometry, mathematical physics, and economics.
Finally, the article by Erik Darpo presents interesting new aspects and results pertaining to the problem of classification of finite-dimensional real division algebras with deep connections to geometry and invariants.