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I wonder if all finite-dimensional algebras over real numbers are isomorphic to an algebra where we add symbols (like $i,j,k$) and define an arbitrary multiplication table for them. For example complex numbers, dual numbers, quaternions, octonions, Clifford algebra etc.

And my other question is whether all such algebras with symbols and arbitrary multiplication tables are finite-dimensional algebras over real numbers.

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Pretty much implicitly that's true, by the definition of "finite-dimensional." – Thomas Andrews Mar 21 '13 at 14:12
Deleting my answer based on the assumption of associativity. Without associativity the structure constants are unconstrained, and Thomas Andrews' comment holds. – Jyrki Lahtonen Mar 21 '13 at 15:09

There is an appropriate theorem:

Bautista, R.; Gabriel, P.; Rojter, A.V.; Salmerón, L. Representation-finite algebras and multiplicative bases. Invent. Math. 81, 217-285 (1985).

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