diff --git a/gtsam/base/Group.h b/gtsam/base/Group.h
index f520b2ff7..f35091757 100644
--- a/gtsam/base/Group.h
+++ b/gtsam/base/Group.h
@@ -128,7 +128,7 @@ compose_pow(const G& g, size_t n) {
 
 /// Template to construct the direct product of two arbitrary groups
 /// Assumes nothing except group structure from G and H
-template<class Derived, typename G, typename H>
+template<typename G, typename H>
 class DirectProduct: public std::pair<G, H> {
   BOOST_CONCEPT_ASSERT((IsGroup<G>));
   BOOST_CONCEPT_ASSERT((IsGroup<H>));
@@ -140,23 +140,23 @@ public:
   // Construct from two subgroup elements
   DirectProduct(const G& g, const H& h):std::pair<G,H>(g,h) {}
 
-  Derived operator*(const Derived& other) const {
-    return Derived(traits<G>::Compose(this->first, other.first),
+  DirectProduct operator*(const DirectProduct& other) const {
+    return DirectProduct(traits<G>::Compose(this->first, other.first),
         traits<H>::Compose(this->second, other.second));
   }
-  Derived inverse() const {
-    return Derived(this->first.inverse(), this->second.inverse());
+  DirectProduct inverse() const {
+    return DirectProduct(this->first.inverse(), this->second.inverse());
   }
 };
 
 // Define any direct product group to be a model of the multiplicative Group concept
-template<class Derived, typename G, typename H>
-struct traits<DirectProduct<Derived, G, H> > :
-  internal::MultiplicativeGroupTraits<DirectProduct<Derived, G, H> > {};
+template<typename G, typename H>
+struct traits<DirectProduct<G, H> > :
+  internal::MultiplicativeGroupTraits<DirectProduct<G, H> > {};
 
 /// Template to construct the direct sum of two additive groups
 /// Assumes existence of three additive operators for both groups
-template<class Derived, typename G, typename H>
+template<typename G, typename H>
 class DirectSum: public std::pair<G, H> {
   BOOST_CONCEPT_ASSERT((IsGroup<G>));  // TODO(frank): check additive
   BOOST_CONCEPT_ASSERT((IsGroup<H>));  // TODO(frank): check additive
@@ -171,21 +171,21 @@ public:
   // Construct from two subgroup elements
   DirectSum(const G& g, const H& h):std::pair<G,H>(g,h) {}
 
-  Derived operator+(const Derived& other) const {
+  DirectSum operator+(const DirectSum& other) const {
     return DirectSum(g()+other.g(), h()+other.h());
   }
-  Derived operator-(const Derived& other) const {
-    return Derived(g()-other.g(), h()-other.h());
+  DirectSum operator-(const DirectSum& other) const {
+    return DirectSum(g()-other.g(), h()-other.h());
   }
-  Derived operator-() const {
-    return Derived(- g(), - h());
+  DirectSum operator-() const {
+    return DirectSum(- g(), - h());
   }
 };
 
 // Define direct sums to be a model of the Additive Group concept
-template<class Derived, typename G, typename H>
-struct traits<DirectSum<Derived, G, H> > :
-  internal::AdditiveGroupTraits<DirectSum<Derived, G, H> > {};
+template<typename G, typename H>
+struct traits<DirectSum<G, H> > :
+  internal::AdditiveGroupTraits<DirectSum<G, H> > {};
 
 }  // namespace gtsam
 
diff --git a/gtsam/base/Manifold.h b/gtsam/base/Manifold.h
index 2f8dc5f68..12df84819 100644
--- a/gtsam/base/Manifold.h
+++ b/gtsam/base/Manifold.h
@@ -170,14 +170,14 @@ struct FixedDimension {
       "FixedDimension instantiated for dymanically-sized type.");
 };
 
-/// CRTP to construct the product manifold of two other manifolds, M1 and M2
-/// Assumes manifold structure from M1 and M2, and binary constructor
-template<class Derived, typename M1, typename M2>
+/// Helper class to construct the product manifold of two other manifolds, M1 and M2
+/// Assumes nothing except manifold structure from M1 and M2
+template<typename M1, typename M2>
 class ProductManifold: public std::pair<M1, M2> {
   BOOST_CONCEPT_ASSERT((IsManifold<M1>));
   BOOST_CONCEPT_ASSERT((IsManifold<M2>));
 
-private:
+protected:
   enum { dimension1 = traits<M1>::dimension };
   enum { dimension2 = traits<M2>::dimension };
 
@@ -196,14 +196,14 @@ public:
   ProductManifold(const M1& m1, const M2& m2):std::pair<M1,M2>(m1,m2) {}
 
   /// Retract delta to manifold
-  Derived retract(const TangentVector& xi) const {
+  ProductManifold retract(const TangentVector& xi) const {
     M1 m1 = traits<M1>::Retract(this->first,  xi.template head<dimension1>());
     M2 m2 = traits<M2>::Retract(this->second, xi.template tail<dimension2>());
-    return Derived(m1,m2);
+    return ProductManifold(m1,m2);
   }
 
   /// Compute the coordinates in the tangent space
-  TangentVector localCoordinates(const Derived& other) const {
+  TangentVector localCoordinates(const ProductManifold& other) const {
     typename traits<M1>::TangentVector v1 = traits<M1>::Local(this->first,  other.first);
     typename traits<M2>::TangentVector v2 = traits<M2>::Local(this->second, other.second);
     TangentVector v;
@@ -213,9 +213,8 @@ public:
 };
 
 // Define any direct product group to be a model of the multiplicative Group concept
-template<class Derived, typename M1, typename M2>
-struct traits<ProductManifold<Derived, M1, M2> > : internal::Manifold<
-    ProductManifold<Derived, M1, M2> > {
+template<typename M1, typename M2>
+struct traits<ProductManifold<M1, M2> > : internal::Manifold<ProductManifold<M1, M2> > {
 };
 
 } // \ namespace gtsam
diff --git a/gtsam/base/tests/testGroup.cpp b/gtsam/base/tests/testGroup.cpp
index 034c7acaf..dadf2896c 100644
--- a/gtsam/base/tests/testGroup.cpp
+++ b/gtsam/base/tests/testGroup.cpp
@@ -103,11 +103,7 @@ TEST(Group, S3) {
 //******************************************************************************
 // The direct product of S2=Z2 and S3 is the symmetry group of a hexagon,
 // i.e., the dihedral group of order 12 (denoted Dih6 because 6-sided polygon)
-struct Dih6 : DirectProduct<Dih6, S2, S3> {
-  typedef DirectProduct<Dih6, S2, S3> Base;
-  Dih6(const S2& g, const S3& h):Base(g,h) {}
-  Dih6() {}
-};
+typedef DirectProduct<S2, S3> Dih6;
 
 std::ostream &operator<<(std::ostream &os, const Dih6& m) {
   os << "( " << m.first << ", " << m.second << ")";
diff --git a/gtsam/geometry/EssentialMatrix.h b/gtsam/geometry/EssentialMatrix.h
index 29675d640..697bd462d 100644
--- a/gtsam/geometry/EssentialMatrix.h
+++ b/gtsam/geometry/EssentialMatrix.h
@@ -21,13 +21,17 @@ namespace gtsam {
  * but here we choose instead to parameterize it as a (Rot3,Unit3) pair.
  * We can then non-linearly optimize immediately on this 5-dimensional manifold.
  */
-class GTSAM_EXPORT EssentialMatrix : private ProductManifold<EssentialMatrix, Rot3, Unit3> {
+class GTSAM_EXPORT EssentialMatrix : private ProductManifold<Rot3, Unit3> {
 
 private:
-  friend class ProductManifold<EssentialMatrix, Rot3, Unit3>;
-  typedef ProductManifold<EssentialMatrix, Rot3, Unit3> Base;
+  typedef ProductManifold<Rot3, Unit3> Base;
   Matrix3 E_; ///< Essential matrix
 
+  /// Construct from Base
+  EssentialMatrix(const Base& base) :
+      Base(base), E_(direction().skew() * rotation().matrix()) {
+  }
+
 public:
 
   /// Static function to convert Point2 to homogeneous coordinates
@@ -82,9 +86,16 @@ public:
   using Base::dimension;
   using Base::dim;
   using Base::Dim;
-  using Base::retract;
-  using Base::localCoordinates;
 
+  /// Retract delta to manifold
+  EssentialMatrix retract(const TangentVector& v) const {
+    return Base::retract(v);
+  }
+
+  /// Compute the coordinates in the tangent space
+  TangentVector localCoordinates(const EssentialMatrix& other) const {
+    return Base::localCoordinates(other);
+  }
   /// @}
 
   /// @name Essential matrix methods
diff --git a/gtsam/geometry/tests/testCyclic.cpp b/gtsam/geometry/tests/testCyclic.cpp
index 84125ef22..7becfc75f 100644
--- a/gtsam/geometry/tests/testCyclic.cpp
+++ b/gtsam/geometry/tests/testCyclic.cpp
@@ -87,12 +87,7 @@ TEST(Cyclic , Invariants) {
 //******************************************************************************
 // The Direct sum of Z2 and Z2 is *not* Cyclic<4>, but the
 // smallest non-cyclic group called the Klein four-group:
-struct K4: DirectSum<K4, Z2, Z2> {
-  typedef DirectSum<K4, Z2, Z2> Base;
-  K4(const Z2& g, const Z2& h):Base(g,h) {}
-  K4(const Base& base):Base(base) {}
-  K4() {}
-};
+typedef DirectSum<Z2, Z2> K4;
 
 namespace gtsam {
 
diff --git a/tests/testManifold.cpp b/tests/testManifold.cpp
index 496579b8d..89b296824 100644
--- a/tests/testManifold.cpp
+++ b/tests/testManifold.cpp
@@ -10,13 +10,14 @@
  * -------------------------------1------------------------------------------- */
 
 /**
- * @file testExpression.cpp
+ * @file testManifold.cpp
  * @date September 18, 2014
  * @author Frank Dellaert
  * @author Paul Furgale
- * @brief unit tests for Block Automatic Differentiation
+ * @brief unit tests for Manifold type machinery
  */
 
+#include <gtsam/base/Manifold.h>
 #include <gtsam/geometry/PinholeCamera.h>
 #include <gtsam/geometry/Pose2.h>
 #include <gtsam/geometry/Cal3_S2.h>
@@ -149,12 +150,7 @@ TEST(Manifold, DefaultChart) {
 }
 
 //******************************************************************************
-struct MyPoint2Pair : public ProductManifold<MyPoint2Pair,Point2,Point2> {
-  typedef ProductManifold<MyPoint2Pair,Point2,Point2> Base;
-  MyPoint2Pair(const Point2& p1, const Point2& p2):Base(p1,p2) {}
-  MyPoint2Pair(const Base& base):Base(base) {}
-  MyPoint2Pair() {}
-};
+typedef ProductManifold<Point2,Point2> MyPoint2Pair;
 
 // Define any direct product group to be a model of the multiplicative Group concept
 namespace gtsam {