PauliSimplify[exp] simplifies products of Pauli matrices and expands non-commutative products. Double indices and vectors are contracted. The order of the Pauli matrices is not changed.
Overview, PauliSigma, PauliTrick.
CSIS[p1] . CSI[i] . CSIS[p2]
PauliSimplify[%]\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)
CSIS[p] . CSI[i, j, k] . CSIS[p]
PauliSimplify[%]\left(\overline{\sigma }\cdot \overline{p}\right).\overline{\sigma }^i.\overline{\sigma }^j.\overline{\sigma }^k.\left(\overline{\sigma }\cdot \overline{p}\right)
-\overline{p}^2 \overline{\sigma }^i.\overline{\sigma }^j.\overline{\sigma }^k+2 \overline{p}^k \overline{\sigma }^i.\overline{\sigma }^j.\left(\overline{\sigma }\cdot \overline{p}\right)-2 \overline{p}^j \overline{\sigma }^i.\overline{\sigma }^k.\left(\overline{\sigma }\cdot \overline{p}\right)+2 \overline{p}^i \overline{\sigma }^j.\overline{\sigma }^k.\left(\overline{\sigma }\cdot \overline{p}\right)
PauliSimplify[CSIS[p] . CSI[i, j, k] . CSIS[p], PauliReduce -> False]-\overline{p}^2 \overline{\sigma }^i.\overline{\sigma }^j.\overline{\sigma }^k+2 \overline{p}^k \overline{\sigma }^i.\overline{\sigma }^j.\left(\overline{\sigma }\cdot \overline{p}\right)-2 \overline{p}^j \overline{\sigma }^i.\overline{\sigma }^k.\left(\overline{\sigma }\cdot \overline{p}\right)+2 \overline{p}^i \overline{\sigma }^j.\overline{\sigma }^k.\left(\overline{\sigma }\cdot \overline{p}\right)
CSID[i, j, i]
PauliSimplify[%]\sigma ^i.\sigma ^j.\sigma ^i
3 \sigma ^j-D \sigma ^j
CSID[i, j, k, l, m, i]
PauliSimplify[%]\sigma ^i.\sigma ^j.\sigma ^k.\sigma ^l.\sigma ^m.\sigma ^i
D \sigma ^j.\sigma ^k.\sigma ^l.\sigma ^m-3 \sigma ^j.\sigma ^k.\sigma ^l.\sigma ^m+2 \sigma ^j.\sigma ^k.\sigma ^m.\sigma ^l-2 \sigma ^j.\sigma ^l.\sigma ^m.\sigma ^k+2 \sigma ^k.\sigma ^l.\sigma ^m.\sigma ^j