GSE[p] can be used as input for a D-4-dimensional \gamma \cdot p = \gamma^\mu p_\mu and is
transformed into DiracGamma[Momentum[p,D-4],D-4] by
FeynCalcInternal (FCI).
GSE[p,q, ...] is a short form for
GSE[p].GSE[q]. ....
Overview, DiracGamma, GA, GAD, GSD.
GSE[p]\hat{\gamma }\cdot \hat{p}
GSE[p] // FCI // StandardForm
(*DiracGamma[Momentum[p, -4 + D], -4 + D]*)GSE[p, q, r, s]\left(\hat{\gamma }\cdot \hat{p}\right).\left(\hat{\gamma }\cdot \hat{q}\right).\left(\hat{\gamma }\cdot \hat{r}\right).\left(\hat{\gamma }\cdot \hat{s}\right)
GSE[p, q, r, s] // StandardForm
(*GSE[p] . GSE[q] . GSE[r] . GSE[s]*)GSE[q] . (GSE[p] + m) . GSE[q]\left(\hat{\gamma }\cdot \hat{q}\right).\left(m+\hat{\gamma }\cdot \hat{p}\right).\left(\hat{\gamma }\cdot \hat{q}\right)
In order to use Dirac algebra with D-4 dimensional objects you need to activate the t’Hooft-Veltman-Breitenlohner-Maison scheme first
FCSetDiracGammaScheme["NDR"];
DiracSimplify[GSE[q] . GS[q] . GSE[q]]
\text{\$Aborted}
FCSetDiracGammaScheme["BMHV"];
DiracSimplify[GSE[q] . GS[q] . GSE[q]]\hat{q}^2 \left(-\left(\bar{\gamma }\cdot \overline{q}\right)\right)
FCSetDiracGammaScheme["NDR"];