PauliChainCombine[exp] is (nearly) the inverse operation
to PauliChainExpand.
Overview, PauliChain, PCHN, PauliIndex, PauliIndexDelta, DIDelta, PauliChainJoin, PauliChainExpand, PauliChainFactor.
(PCHN[CSISD[q], Dir3, Dir4] FAD[{k, me}])/(2 CSPD[q, q]) + 1/(2 CSPD[q, q])*
FAD[k, {k - q, me}] (-2 DCHN[CSISD[q], Dir3, Dir4] CSPD[q, q] + 2 DCHN[1, Dir3, Dir4]*
me CSPD[q, q] + DCHN[CSISD[q], Dir3, Dir4] (-me^2 + CSPD[q, q]))
PauliChainCombine[%]\frac{\left(q^2-\text{me}^2\right) \left((1)_{\text{Dir3}\;\text{Dir4}} \sigma \cdot q\right)+2 \;\text{me} q^2 (1)_{\text{Dir3}\;\text{Dir4}}-2 q^2 \left((1)_{\text{Dir3}\;\text{Dir4}} \sigma \cdot q\right)}{2 q^2 k^2.\left((k-q)^2-\text{me}^2\right)}+\frac{(\sigma \cdot q)_{\text{Dir3}\;\text{Dir4}}}{2 q^2 \left(k^2-\text{me}^2\right)}
\frac{(1)_{\text{Dir3}\;\text{Dir4}} \left(q^2-\text{me}^2\right) \sigma \cdot q+2 \;\text{me} q^2 (1)_{\text{Dir3}\;\text{Dir4}}-2 q^2 (1)_{\text{Dir3}\;\text{Dir4}} \sigma \cdot q}{2 q^2 k^2.\left((k-q)^2-\text{me}^2\right)}+\frac{(\sigma \cdot q)_{\text{Dir3}\;\text{Dir4}}}{2 q^2 \left(k^2-\text{me}^2\right)}