PauliTrick[exp] contracts \sigma matrices with each other and performs several simplifications (no expansion, use PauliSimplify for this).
Overview, PauliSigma, PauliSimplify.
CSIS[p1] . CSI[i] . CSIS[p2]
PauliTrick[%] // Contract\left(\overline{\sigma }\cdot \overline{\text{p1}}\right).\overline{\sigma }^i.\left(\overline{\sigma }\cdot \overline{\text{p2}}\right)
\left(\overline{\sigma }\cdot \overline{\text{p1}}\right).\overline{\sigma }^i.\left(\overline{\sigma }\cdot \overline{\text{p2}}\right)
CSID[i, j, i]
PauliTrick[%] // Contract\sigma ^i.\sigma ^j.\sigma ^i
-\left((D-3) \sigma ^j\right)
CSIS[p] . CSI[j] . CSIS[p] . CSIS[i]
PauliTrick[%] // Contract // EpsEvaluate // FCCanonicalizeDummyIndices
PauliTrick[%%, PauliReduce -> False]\left(\overline{\sigma }\cdot \overline{p}\right).\overline{\sigma }^j.\left(\overline{\sigma }\cdot \overline{p}\right).\left(\overline{\sigma }\cdot \overline{i}\right)
2 \overline{p}^j \left(\overline{\sigma }\cdot \overline{p}\right).\left(\overline{\sigma }\cdot \overline{i}\right)-\overline{p}^2 \overline{\sigma }^j.\left(\overline{\sigma }\cdot \overline{i}\right)
2 \overline{p}^j \left(\overline{\sigma }\cdot \overline{p}\right).\left(\overline{\sigma }\cdot \overline{i}\right)-\overline{p}^2 \overline{\sigma }^j.\left(\overline{\sigma }\cdot \overline{i}\right)