  In this paper, we will define analogues of multiple zeta values by replacing
the differential forms defining multiple zeta values with some
$\mathbb{Q}$-rational differential forms on the Fermat curve $F_2$ of degree 2
and discuss their arithmetic properties. We also investigate a motivic
structure of the motivic periods corresponding to our periods. However, in
order to study them, the current theory for motivic zeta elements is
insufficient, and it leads us to study the base extension of the space of the
motivic periods $\mathcal{H}_4$ of level 4 and its Galois invariant part.
