diff --git a/gtsam/base/Group.h b/gtsam/base/Group.h
index f35091757..39541416e 100644
--- a/gtsam/base/Group.h
+++ b/gtsam/base/Group.h
@@ -20,6 +20,8 @@
 
 #pragma once
 
+#include <gtsam/base/Testable.h>
+
 #include <boost/concept_check.hpp>
 #include <boost/concept/requires.hpp>
 #include <boost/type_traits/is_base_of.hpp>
@@ -87,34 +89,38 @@ check_group_invariants(const G& a, const G& b, double tol = 1e-9) {
 
 namespace internal {
 
-/// A helper class that implements the traits interface for groups.
-/// Assumes that constructor yields identity
+/// A helper class that implements the traits interface for multiplicative groups.
+/// Assumes existence of identity, operator* and inverse method
 template<class Class>
-struct GroupTraits {
+struct MultiplicativeGroupTraits {
   typedef group_tag structure_category;
-  static Class Identity() { return Class(); }
+  typedef multiplicative_group_tag group_flavor;
+  static Class Identity() { return Class::identity(); }
+  static Class Compose(const Class &g, const Class & h) { return g * h;}
+  static Class Between(const Class &g, const Class & h) { return g.inverse() * h;}
+  static Class Inverse(const Class &g) { return g.inverse();}
 };
 
-/// A helper class that implements the traits interface for multiplicative groups.
-/// Assumes existence of operator* and inverse method
+/// Both multiplicative group traits and Testable
 template<class Class>
-struct MultiplicativeGroupTraits : GroupTraits<Class> {
-    typedef multiplicative_group_tag group_flavor; \
-    static Class Compose(const Class &g, const Class & h) { return g * h;} \
-    static Class Between(const Class &g, const Class & h) { return g.inverse() * h;} \
-    static Class Inverse(const Class &g) { return g.inverse();}
-};
+struct MultiplicativeGroup : MultiplicativeGroupTraits<Class>, Testable<Class> {};
 
 /// A helper class that implements the traits interface for additive groups.
-/// Assumes existence of three additive operators
+/// Assumes existence of identity and three additive operators
 template<class Class>
-struct AdditiveGroupTraits : GroupTraits<Class> {
-    typedef additive_group_tag group_flavor; \
-    static Class Compose(const Class &g, const Class & h) { return g + h;} \
-    static Class Between(const Class &g, const Class & h) { return h - g;} \
-    static Class Inverse(const Class &g) { return -g;}
+struct AdditiveGroupTraits {
+  typedef group_tag structure_category;
+  typedef additive_group_tag group_flavor;
+  static Class Identity() { return Class::identity(); }
+  static Class Compose(const Class &g, const Class & h) { return g + h;}
+  static Class Between(const Class &g, const Class & h) { return h - g;}
+  static Class Inverse(const Class &g) { return -g;}
 };
 
+/// Both additive group traits and Testable
+template<class Class>
+struct AdditiveGroup : AdditiveGroupTraits<Class>, Testable<Class> {};
+
 }  // namespace internal
 
 /// compose multiple times
@@ -127,7 +133,7 @@ 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
+/// Assumes nothing except group structure and Testable from G and H
 template<typename G, typename H>
 class DirectProduct: public std::pair<G, H> {
   BOOST_CONCEPT_ASSERT((IsGroup<G>));
@@ -140,6 +146,9 @@ public:
   // Construct from two subgroup elements
   DirectProduct(const G& g, const H& h):std::pair<G,H>(g,h) {}
 
+  // identity
+  static DirectProduct identity() { return DirectProduct(); }
+
   DirectProduct operator*(const DirectProduct& other) const {
     return DirectProduct(traits<G>::Compose(this->first, other.first),
         traits<H>::Compose(this->second, other.second));
@@ -171,6 +180,9 @@ public:
   // Construct from two subgroup elements
   DirectSum(const G& g, const H& h):std::pair<G,H>(g,h) {}
 
+  // identity
+  static DirectSum identity() { return DirectSum(); }
+
   DirectSum operator+(const DirectSum& other) const {
     return DirectSum(g()+other.g(), h()+other.h());
   }
diff --git a/gtsam/base/Lie.h b/gtsam/base/Lie.h
index c9f788992..05c7bc20f 100644
--- a/gtsam/base/Lie.h
+++ b/gtsam/base/Lie.h
@@ -136,8 +136,10 @@ namespace internal {
 /// A helper class that implements the traits interface for GTSAM lie groups.
 /// To use this for your gtsam type, define:
 /// template<> struct traits<Class> : public internal::LieGroupTraits<Class> {};
+/// Assumes existence of: identity, dimension, localCoordinates, retract,
+/// and additionally Logmap, Expmap, compose, between, and inverse
 template<class Class>
-struct LieGroupTraits : Testable<Class> {
+struct LieGroupTraits {
   typedef lie_group_tag structure_category;
 
   /// @name Group
@@ -167,12 +169,10 @@ struct LieGroupTraits : Testable<Class> {
       ChartJacobian Horigin = boost::none, ChartJacobian Hv = boost::none) {
     return origin.retract(v, Horigin, Hv);
   }
-
   /// @}
 
   /// @name Lie Group
   /// @{
-
   static TangentVector Logmap(const Class& m, ChartJacobian Hm = boost::none) {
     return Class::Logmap(m, Hm);
   }
@@ -195,11 +195,12 @@ struct LieGroupTraits : Testable<Class> {
       ChartJacobian H = boost::none) {
     return m.inverse(H);
   }
-
   /// @}
-
 };
 
+/// Both LieGroupTraits and Testable
+template<class Class> struct LieGroup: LieGroupTraits<Class>, Testable<Class> {};
+
 } // \ namepsace internal
 
 /**
diff --git a/gtsam/base/Manifold.h b/gtsam/base/Manifold.h
index 12df84819..b20c60822 100644
--- a/gtsam/base/Manifold.h
+++ b/gtsam/base/Manifold.h
@@ -116,10 +116,8 @@ struct ManifoldTraits: ManifoldImpl<Class, Class::dimension> {
   }
 };
 
-/// Implement both manifold and testable traits at the same time
-template<class Class>
-struct Manifold: Testable<Class>, ManifoldTraits<Class> {
-};
+/// Both ManifoldTraits and Testable
+template<class Class> struct Manifold: ManifoldTraits<Class>, Testable<Class> {};
 
 } // \ namespace internal
 
diff --git a/gtsam/base/VectorSpace.h b/gtsam/base/VectorSpace.h
index 558fe52c9..c356dcc07 100644
--- a/gtsam/base/VectorSpace.h
+++ b/gtsam/base/VectorSpace.h
@@ -20,7 +20,7 @@ template<typename T> struct traits;
 
 namespace internal {
 
-/// VectorSpace Implementation for Fixed sizes
+/// VectorSpaceTraits Implementation for Fixed sizes
 template<class Class, int N>
 struct VectorSpaceImpl {
 
@@ -83,7 +83,7 @@ struct VectorSpaceImpl {
   /// @}
 };
 
-/// VectorSpace implementation for dynamic types.
+/// VectorSpaceTraits implementation for dynamic types.
 template<class Class>
 struct VectorSpaceImpl<Class,Eigen::Dynamic> {
 
@@ -159,11 +159,11 @@ struct VectorSpaceImpl<Class,Eigen::Dynamic> {
   /// @}
 };
 
-/// A helper that implements the traits interface for GTSAM lie groups.
+/// A helper that implements the traits interface for GTSAM vector space types.
 /// To use this for your gtsam type, define:
-/// template<> struct traits<Type> : public VectorSpace<Type> { };
+/// template<> struct traits<Type> : public VectorSpaceTraits<Type> { };
 template<class Class>
-struct VectorSpace: Testable<Class>, VectorSpaceImpl<Class, Class::dimension> {
+struct VectorSpaceTraits: VectorSpaceImpl<Class, Class::dimension> {
 
   typedef vector_space_tag structure_category;
 
@@ -178,9 +178,12 @@ struct VectorSpace: Testable<Class>, VectorSpaceImpl<Class, Class::dimension> {
   enum { dimension = Class::dimension};
   typedef Class ManifoldType;
   /// @}
-
 };
 
+/// Both VectorSpaceTRaits and Testable
+template<class Class>
+struct VectorSpace: Testable<Class>, VectorSpaceTraits<Class> {};
+
 /// A helper that implements the traits interface for GTSAM lie groups.
 /// To use this for your gtsam type, define:
 /// template<> struct traits<Type> : public ScalarTraits<Type> { };