Composition algebras

number that can be put in the form a + bi, where a and b are real numbers and i is called the imaginary unit

{| class="wikitable" align="right" style="text-align:center; margin-left:0.5em; max-width: 230px;" |+ Quaternion multiplication table |- |width=15| !width=15| !width=15| !width=15| !width=15| |- ! | | | | |- ! | | | | |- ! | | | | |- ! | | | | |- |colspan=5| Left column shows the left factor, top row shows the right factor. Also, a\mathbf{b}=\mathbf{b}a and -\mathbf{b} = (-1)\mathbf{b} for a\in \mathbb{R} , \mathbf{b} = \mathbf{i}, \mathbf{j}, \mathbf{k} . |} thumb|Cayley graph of the [[quaternion group showing the six cycles of multiplication by , and . (If the image

In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.

element of the real commutative associative algebra ℝ[j] / (j² − 1), i.e. the reals with an extra square root of +1 adjoined

manner in which higher-dimension algebras may be produced

commutative, associative algebra of two complex dimensions

In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: Biquaternions when the coefficients are complex numbers. Split-biquaternions when the coefficients are split-complex numbers. Dual quaternions when the coefficients are dual numbers.

theorem on finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form

not necessarily associative algebra

{|class="wikitable" align="right" style="text-align:center" |+Split-quaternion multiplication |- !width=15| × !width=15| 1 !width=15| i !width=15| j !width=15| k |- ! 1 | 1 | i | j | k |- !i |i |−1 |k |−j |- !j |j |−k |1 |−i |- !k |k |j |i |1 |}

generalization of quaternions to other fields

In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signatures of their quadratic forms differ: the split-octonions have a split signature (4,4) whereas the octonions have a positive-definite signature (8,0).