# On Reduced Exceptional Simple Jordan Algebras

• A. A. Albert
• N. Jacobson
Part of the Contemporary Mathematicians book series (CM)

## Abstract

In this paper we shall be concerned only with finite dimensional algebras over an arbitrary field $$\mathfrak{F}$$ of characteristic not two. Let $$\mathfrak{A}$$ be an associative algebra over $$\mathfrak{F}$$ and ab the associative product composition of $$\mathfrak{A}$$. Then the vector space $$\mathfrak{A}$$ is a Jordan algebra $${\mathfrak{A}^{\left( + \right)}}$$ relative to the composition a·b = 1/2(ab + ba), that is, this composition satisfies the defining identities
$$a \cdot b = b \cdot a,\,\left[ {\left( {a \cdot a} \right) \cdot } \right] \cdot a = \left( {a \cdot a} \right) \cdot \left( {b \cdot a} \right)$$
(1)
The algebra $${\mathfrak{A}^{\left( + \right)}}$$ and its subalgebras are called special Jordan algebras.

## Keywords

Division Algebra Jordan Algebra Witt Index Composition Algebra Algebraic Number Field
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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