Formal finite multiple zeta values

We study the algebra of formal finite multiple zeta values by imposing the stuffle and linear shuffle relations of Kaneko and Zagier. Our first results are a surjective homomorphism to the algebra of formal symmetric multiple zeta values and a parity reduction in depth at most four. In even weight we construct a quotient of the formal double zeta space which surjects onto the space of finite multiple zeta values of depth at most four, and we attach to every even period polynomial an explicit relation among the values ${\zeta }_A(2a,1,2b,1)$.