, this may be replaced by what is then an equivalent requirement, where the bilinear form may additionally be restricted to being symmetric without loss of generality. A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains , namely the tensor algebra , and then enforce the fundamental identity by taking a suitable quotient. In our case we want to take the two-sided ideal in generated by all elements of the form for all and define as the quotient algebra The ring product inherited by this quotient is sometimes referred to as the Clifford product to distinguish it from the exterior product and the scalar product. It is then straightforward to show that contains and satisfies the above universal property, so that is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra . It also follows from this construction that is injective. One usually drops the and considers as a linear subspace of . The universal characterization of the Clifford algebra shows that the construction of is in nature. Namely, can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps preserving the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (preserving the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras. Basis and dimension Since comes equipped with a quadratic form , in characteristic not equal to there exist bases for that are orthogonal. An orthogonal basis is one such that for a symmetric bilinear form for , and The fundamental Clifford identity implies that for an orthogonal basis for , and This makes manipulation of orthogonal basis vectors quite simple. Given a product of distinct orthogonal basis vectors of , one can put them into a standard order while including an overall sign determined by the number of pairwise swaps needed to do so (i.e. the signature of the ordering permutation). If the dimension of over is and is an orthogonal basis of , then is free over with a basis The empty product () is defined as the multiplicative identity element. For each value of there are basis elements, so the total dimension of the Clifford algebra is Examples: real and complex Clifford algebras The most important Clifford algebras