Quaternionic covers and monodromy of the Kontsevich-Zorich cocycle in orthogonal groups

We give an example of a Teichmüller curve which contains, in a factor of its monodromy, a group which was not observed before. Namely, it has Zariski closure equal to the group SO^∗(6) in its standard representation; up to finite index, this is the same as SU(3, 1) in its second exterior power representation. The example is constructed using origamis (i.e. square-tiled surfaces). It can be generalized to give monodromy inside the group SO^∗(2n) for all n, but in the general case the monodromy might split further inside the group. Also, we take the opportunity to compute the multiplicities of representations in the (0, 1) part of the cohomology of regular origamis, answering a question of Matheus-Yoccoz-Zmiaikou.