The resulting algebra is again a division algebra, the octonions O, constructed independently by Graves in 1843 and Cayley in 1845.
Although the octonions fail to be associative, they satisfy the weaker condition of alternativity, meaning that any subalgebra generated by two elements is associative.
8] of 3 x 3 Hermitian matrices over the Octonions
is simple with Pierce constant d = 8 but does not admit a representation.
The importance of these identities is that they are intimately connected to certain division algebras, namely the ones for the reals, complex numbers, quaternions, and octonions
, corresponding to dimension n = 1, 2, 4, 8, .
The first set of 120 quaternions can be expressed as 120 special unit quaternions known as icosians which telescope 4-D discrete-space quaternions up to being 8-D discrete-space octonions to locate points that form a special lattice in [R.
4] even though we can now define identical quaternion operations via octonions in the much larger [R.
Together, these two D8 lattices of 120 icosians each combine to form the 240 octonions that define the famous [E.
8] groups results in a subgroup of the continuous group PSL(2,O), where O represents all the unit octonions.
These special 120 icosians are to be considered as special octonions, 8-tuples of rational numbers which, with respect to a particular norm, form part of a special lattice in [R.
The 120 unit quaternions reciprocal to the ones above will meet this requirement as well as define an equivalent set for the reciprocal hypericosahedron, and this second set of 120 octonions also forms part of a special lattice in [R.
Again, there are the same 240 icosian connections to octonions in [R.
Each finite group of octonions acts as rotations and as vectors in [R.