What is an even Clifford algebra? I'm reading a paper and he defines $C_0(f)$ to be the "even Clifford algebra over $R$ associated to $f$", where $R$ is a principal ideal domain and $f$ is a non-degenerate ternary quadratic form. What is meant by an 'even' Clifford algebra?
 A: This is from Cassels, Rational Quadratic Forms, chapter 10, especially pages 177-178. We have a basis $e_1, e_2, e_3$ of a vector space $V$ of dimension 3 over a field.  These satisfy
$$  e_i e_i = f(e_i) $$ and
$$ e_i e_j + e_j e_i =0, \; \mbox{when} \; i \neq j,   $$ meaning that they are orthogonal. Then a basis of the even Clifford algebra is
 $$ 1, \; e_2 e_3, \; e_3 e_1, \; e_1 e_2.$$
You can work out things for a PID. 
Note that Lemurell uses lower case letter $e$ where Cassels uses upper case, in
 $$  E_1 = e_2 e_3, \; E_2 = e_3 e_1, \; E_3 = e_1 e_2.$$
A: Perhaps Wikipedia it...
Clifford Algebras from what I recall are algebraic extensions of the complex plane. ex: The Quaternions.
In C: a Number N = a + bi
In Q (Quaternions): a Number N = a + bi + cj + dk
This is a valid algebra where the following identities:
i^2 = -1, j^2 = -1, and k^2 = -1 ---> ijk = -1 
hold.
Even may refer to the number of terms in the algebra (ex: complex have 2 terms, Quaternions have 4, etc...) or it might be more subtle than that. I can't answer that part but it is best you start looking up the terms:
Complex Numbers,
Quaternions,
Octonion,
Split-Complex Numbers
and from there you'll know where to go
