diff --git a/Grid/qcd/action/gauge/GaugeImplTypes.h b/Grid/qcd/action/gauge/GaugeImplTypes.h
index c3e42f80..1cc5ef05 100644
--- a/Grid/qcd/action/gauge/GaugeImplTypes.h
+++ b/Grid/qcd/action/gauge/GaugeImplTypes.h
@@ -145,7 +145,7 @@ public:
     accelerator_for(ss, P.Grid()->oSites(),1,{
       for (int mu = 0; mu < Nd; mu++) {
           U_v[ss](mu) = Exponentiate(P_v[ss](mu), ep, Nexp) * U_v[ss](mu);
-          Group::ProjectOnGaugeGroup(U_v[ss](mu));
+          Group::ProjectOnGeneralGroup(U_v[ss](mu));
       }
     });
    //auto end = std::chrono::high_resolution_clock::now();
@@ -166,7 +166,7 @@ public:
 
   static inline void Project(Field &U)
   {
-      Group::ProjectGn(U);
+      Group::ProjectOnSpecialGroup(U);
   }
 
   static inline void HotConfiguration(GridParallelRNG &pRNG, Field &U)
diff --git a/Grid/qcd/utils/GaugeGroup.h b/Grid/qcd/utils/GaugeGroup.h
index 5a8f51c2..825881e2 100644
--- a/Grid/qcd/utils/GaugeGroup.h
+++ b/Grid/qcd/utils/GaugeGroup.h
@@ -323,37 +323,37 @@ class GaugeGroup {
     }
   }
     
-  template <int N>        // reunitarise, resimplectify...
-  static void ProjectOnGaugeGroup(Lattice<iVector<iScalar<iMatrix<vComplexD, N> >, Nd> > &U) {
+  template <int N>                  // Projects on the general groups U(N), Sp(2N)xZ2 i.e. determinant il allowed a complex phase.
+  static void ProjectOnGeneralGroup(Lattice<iVector<iScalar<iMatrix<vComplexD, N> >, Nd> > &U) {
     for (int mu = 0; mu < Nd; mu++) {
       auto Umu = PeekIndex<LorentzIndex>(U, mu);
-      ProjectOnGaugeGroup(Umu);
+      ProjectOnGeneralGroup(Umu);
     }
   }
        
-  template <int N>        // reunitarise, resimplectify...
-  static void ProjectOnGaugeGroup(Lattice<iScalar<iScalar<iMatrix<vComplexD, N> > > > &Umu) {
-      ProjectOnGaugeGroup(Umu, group_name());
+  template <int N>
+  static void ProjectOnGeneralGroup(Lattice<iScalar<iScalar<iMatrix<vComplexD, N> > > > &Umu) {
+      ProjectOnGeneralGroup(Umu, group_name());
   }
     
   template <class vtype>
-  static void ProjectOnGaugeGroup(iScalar<vtype> &r) {
-      ProjectOnGaugeGroup(r, group_name());
+  static void ProjectOnGeneralGroup(iScalar<vtype> &r) {
+      ProjectOnGeneralGroup(r, group_name());
   }
 
   template <class vtype, int N>
-  static void ProjectOnGaugeGroup(iVector<vtype,N> &r) {
-      ProjectOnGaugeGroup(r, group_name());
+  static void ProjectOnGeneralGroup(iVector<vtype,N> &r) {
+      ProjectOnGeneralGroup(r, group_name());
   }
 
   template <class vtype,int N, typename std::enable_if< GridTypeMapper<vtype>::TensorLevel == 0 >::type * =nullptr>
-  static void ProjectOnGaugeGroup(iMatrix<vtype,N> &arg) {
-      ProjectOnGaugeGroup(arg, group_name());
+  static void ProjectOnGeneralGroup(iMatrix<vtype,N> &arg) {
+      ProjectOnGeneralGroup(arg, group_name());
   }
 
-  template <int N>       // reunitarise, resimplectify... previously ProjectSUn
-  static void ProjectGn(Lattice<iScalar<iScalar<iMatrix<vComplexD, N> > > > &Umu) {
-       ProjectOnGaugeGroup(Umu);
+  template <int N>       // Projects on SU(N), Sp(2N)
+  static void ProjectOnSpecialGroup(Lattice<iScalar<iScalar<iMatrix<vComplexD, N> > > > &Umu) {
+       ProjectOnGeneralGroup(Umu);
        auto det = Determinant(Umu);
 
        det = conjugate(det);
@@ -366,11 +366,11 @@ class GaugeGroup {
    }
 
     template <int N>    // reunitarise, resimplectify... previously ProjectSUn
-    static void ProjectGn(Lattice<iVector<iScalar<iMatrix<vComplexD, N> >, Nd> > &U) {
+    static void ProjectOnSpecialGroup(Lattice<iVector<iScalar<iMatrix<vComplexD, N> >, Nd> > &U) {
       // Reunitarise
       for (int mu = 0; mu < Nd; mu++) {
         auto Umu = PeekIndex<LorentzIndex>(U, mu);
-        ProjectGn(Umu);
+        ProjectOnSpecialGroup(Umu);
         PokeIndex<LorentzIndex>(U, Umu, mu);
       }
     }
diff --git a/Grid/qcd/utils/SUn.impl b/Grid/qcd/utils/SUn.impl
index 38337305..07bfc6e7 100644
--- a/Grid/qcd/utils/SUn.impl
+++ b/Grid/qcd/utils/SUn.impl
@@ -509,22 +509,22 @@ static void testGenerators(GroupName::SU) {
 }
 
 template <int N>
-static void ProjectOnGaugeGroup(Lattice<iScalar<iScalar<iMatrix<vComplexD, N> > > > &Umu, GroupName::SU) {
+static void ProjectOnGeneralGroup(Lattice<iScalar<iScalar<iMatrix<vComplexD, N> > > > &Umu, GroupName::SU) {
   Umu = ProjectOnGroup(Umu);
 }
 
 template <class vtype>
-static void ProjectOnGaugeGroup(iScalar<vtype> &r, GroupName::SU) {
+static void ProjectOnGeneralGroup(iScalar<vtype> &r, GroupName::SU) {
   r = ProjectOnGroup(r);
 }
 
 template <class vtype, int N>
-static void ProjectOnGaugeGroup(iVector<vtype,N> &r, GroupName::SU) {
+static void ProjectOnGeneralGroup(iVector<vtype,N> &r, GroupName::SU) {
   r = ProjectOnGroup(r);
 }
 
 template <class vtype,int N, typename std::enable_if< GridTypeMapper<vtype>::TensorLevel == 0 >::type * =nullptr>
-static void ProjectOnGaugeGroup(iMatrix<vtype,N> &arg, GroupName::SU) {
+static void ProjectOnGeneralGroup(iMatrix<vtype,N> &arg, GroupName::SU) {
   arg = ProjectOnGroup(arg);
 }
 
diff --git a/Grid/qcd/utils/Sp2n.impl b/Grid/qcd/utils/Sp2n.impl
index 2ffb79ad..9cca5d49 100644
--- a/Grid/qcd/utils/Sp2n.impl
+++ b/Grid/qcd/utils/Sp2n.impl
@@ -254,22 +254,22 @@ static void testGenerators(GroupName::Sp) {
 }
 
 template <int N>
-static void ProjectOnGaugeGroup(Lattice<iScalar<iScalar<iMatrix<vComplexD, N> > > > &Umu, GroupName::Sp) {
+static void ProjectOnGeneralGroup(Lattice<iScalar<iScalar<iMatrix<vComplexD, N> > > > &Umu, GroupName::Sp) {
   Umu = ProjectOnSpGroup(Umu);
 }
 
 template <class vtype>
-static void ProjectOnGaugeGroup(iScalar<vtype> &r, GroupName::Sp) {
+static void ProjectOnGeneralGroup(iScalar<vtype> &r, GroupName::Sp) {
   r = ProjectOnSpGroup(r);
 }
 
 template <class vtype, int N>
-static void ProjectOnGaugeGroup(iVector<vtype,N> &r, GroupName::Sp) {
+static void ProjectOnGeneralGroup(iVector<vtype,N> &r, GroupName::Sp) {
   r = ProjectOnSpGroup(r);
 }
 
 template <class vtype,int N, typename std::enable_if< GridTypeMapper<vtype>::TensorLevel == 0 >::type * =nullptr>
-static void ProjectOnGaugeGroup(iMatrix<vtype,N> &arg, GroupName::Sp) {
+static void ProjectOnGeneralGroup(iMatrix<vtype,N> &arg, GroupName::Sp) {
   arg = ProjectOnSpGroup(arg);
 }
 
diff --git a/tests/sp2n/Test_project_on_Sp.cc b/tests/sp2n/Test_project_on_Sp.cc
index 9f42610f..7737bc09 100644
--- a/tests/sp2n/Test_project_on_Sp.cc
+++ b/tests/sp2n/Test_project_on_Sp.cc
@@ -88,20 +88,20 @@ void test_group_projections(T U) {
   U = ProjectOnSpGroup(U);
   assert(is_element_of_sp2n_group(U));
 
-  name = "ProjectOnGaugeGroup";
+  name = "ProjectOnGeneralGroup";
   std::cout << GridLogMessage << "Testing " << name << std::endl;
   std::cout << GridLogMessage << "Apply to deformed matrix" << std::endl;
 
   U = U + Delta * identity;
-  Sp<Nc>::ProjectOnGaugeGroup(U);
+  Sp<Nc>::ProjectOnGeneralGroup(U);
   assert(is_element_of_sp2n_group(U));
 
-  name = "ProjectGn";
+  name = "ProjectOnSpecialGroup";
   std::cout << GridLogMessage << "Testing " << name << std::endl;
   std::cout << GridLogMessage << "Apply to deformed matrix" << std::endl;
 
   U = U + Delta * identity;
-  Sp<Nc>::ProjectGn(U);
+  Sp<Nc>::ProjectOnSpecialGroup(U);
   assert(is_element_of_sp2n_group(U));
 }
 
@@ -213,11 +213,10 @@ int main(int argc, char** argv) {
   LatticeColourMatrixD U(&Grid);
 
   // Will test resimplectification-related functionalities (from
-  // ProjectOnGaugeGroup, ProjectOnSpGroup, ProjectGn) and projection on the
-  // algebra (from ProjectSp2nAlgebra) we work with matrices with positive
-  // determinant so detU = 1 even if in principle ProjectOnGaugeGroup and
-  // ProjectOnSpGroup allow for detU=-1 so the checks will be the same for the
-  // three functions NB only ProjectGn is the proper simplectification function
+  // ProjectOnGeneralGroup, ProjectOnSpGroup, ProjectOnSpecialGroup) and projection on the
+  // algebra (from SpTa)
+  // ProjectOnGeneralGroup, ProjectOnSpGroup project on the non-special group allowi for complex determinants of module 1
+  // ProjectOnSpecialGroup projects on the full gauge group providing a determinant equals to 1
 
   std::vector<int> pseeds({1, 2, 3, 4, 5});
   GridParallelRNG pRNG(&Grid);