diff --git a/gtsam/geometry/Pose2.cpp b/gtsam/geometry/Pose2.cpp
index bb5f8ed79..b5c426597 100644
--- a/gtsam/geometry/Pose2.cpp
+++ b/gtsam/geometry/Pose2.cpp
@@ -204,8 +204,8 @@ Pose2 Pose2::inverse() const {
 }
 
 /* ************************************************************************* */
-Pose2::LieAlgebra Pose2::Hat(const Pose2::TangentVector& xi) {
-  LieAlgebra X;
+Matrix3 Pose2::Hat(const Pose2::TangentVector& xi) {
+  Matrix3 X;
   X << 0., -xi.z(), xi.x(),
     xi.z(), 0., xi.y(),
     0., 0., 0.;
@@ -213,7 +213,7 @@ Pose2::LieAlgebra Pose2::Hat(const Pose2::TangentVector& xi) {
 }
 
 /* ************************************************************************* */
-Pose2::TangentVector Pose2::Vee(const Pose2::LieAlgebra& X) {
+Pose2::TangentVector Pose2::Vee(const Matrix3& X) {
   return TangentVector(X(0, 2), X(1, 2), X(1,0));
 }
 
diff --git a/gtsam/geometry/Pose2.h b/gtsam/geometry/Pose2.h
index 6c7644123..93bc6751b 100644
--- a/gtsam/geometry/Pose2.h
+++ b/gtsam/geometry/Pose2.h
@@ -201,10 +201,10 @@ public:
   using LieGroup<Pose2, 3>::inverse; // version with derivative
 
   /// Hat maps from tangent vector to Lie algebra
-  static LieAlgebra Hat(const TangentVector& xi);
+  static Matrix3 Hat(const Vector3& xi);
 
   /// Vee maps from Lie algebra to tangent vector
-  static TangentVector Vee(const LieAlgebra& X);
+  static Vector3 Vee(const Matrix3& X);
 
   /// @}
   /// @name Group Action on Point2
@@ -338,7 +338,7 @@ public:
 
 #ifdef GTSAM_ALLOW_DEPRECATED_SINCE_V43
   /// @deprecated: use Hat
-  static inline LieAlgebra wedge(double vx, double vy, double w) {
+  static inline Matrix3 wedge(double vx, double vy, double w) {
     return Hat(TangentVector(vx, vy, w));
   }
 #endif
@@ -375,10 +375,10 @@ using Pose2Pair = std::pair<Pose2, Pose2>;
 using Pose2Pairs = std::vector<Pose2Pair>;
 
 template <>
-struct traits<Pose2> : public internal::LieGroup<Pose2> {};
+struct traits<Pose2> : public internal::MatrixLieGroup<Pose2> {};
 
 template <>
-struct traits<const Pose2> : public internal::LieGroup<Pose2> {};
+struct traits<const Pose2> : public internal::MatrixLieGroup<Pose2> {};
 
 // bearing and range traits, used in RangeFactor
 template <typename T>
diff --git a/gtsam/geometry/Pose3.h b/gtsam/geometry/Pose3.h
index 28f7817e6..154e24a36 100644
--- a/gtsam/geometry/Pose3.h
+++ b/gtsam/geometry/Pose3.h
@@ -238,10 +238,10 @@ public:
    *  v (vx,vy,vz) = 3D velocity
    * @return xihat, 4*4 element of Lie algebra that can be exponentiated
    */
-  static LieAlgebra Hat(const TangentVector& xi);
+  static Matrix4 Hat(const Vector6& xi);
 
   /// Vee maps from Lie algebra to tangent vector
-  static TangentVector Vee(const LieAlgebra& X);
+  static Vector6 Vee(const Matrix4& X);
 
   /// @}
   /// @name Group Action on Point3
@@ -446,10 +446,10 @@ using Pose3Pairs = std::vector<std::pair<Pose3, Pose3> >;
 typedef std::vector<Pose3> Pose3Vector;
 
 template <>
-struct traits<Pose3> : public internal::LieGroup<Pose3> {};
+struct traits<Pose3> : public internal::MatrixLieGroup<Pose3> {};
 
 template <>
-struct traits<const Pose3> : public internal::LieGroup<Pose3> {};
+struct traits<const Pose3> : public internal::MatrixLieGroup<Pose3> {};
 
 // bearing and range traits, used in RangeFactor
 template <>
diff --git a/gtsam/geometry/Quaternion.h b/gtsam/geometry/Quaternion.h
index 86fc366ec..f0deba5a9 100644
--- a/gtsam/geometry/Quaternion.h
+++ b/gtsam/geometry/Quaternion.h
@@ -140,11 +140,11 @@ struct traits<QUATERNION_TYPE> {
 
   using LieAlgebra = Matrix3;
 
-  static LieAlgebra Hat(const Vector3& v) {
+  static Matrix3 Hat(const Vector3& v) {
     return SO3::Hat(v);
   }
   
-  static Vector3 Vee(const LieAlgebra& X) {
+  static Vector3 Vee(const Matrix3& X) {
     return SO3::Vee(X);
   }
 
diff --git a/gtsam/geometry/Rot2.cpp b/gtsam/geometry/Rot2.cpp
index 9b57740a3..f1eca66d8 100644
--- a/gtsam/geometry/Rot2.cpp
+++ b/gtsam/geometry/Rot2.cpp
@@ -82,15 +82,15 @@ Vector1 Rot2::Logmap(const Rot2& r, OptionalJacobian<1, 1> H) {
   return v;
 }
 /* ************************************************************************* */
-Rot2::LieAlgebra Rot2::Hat(const Rot2::TangentVector& xi) {
-  LieAlgebra X;
+Matrix2 Rot2::Hat(const Vector1& xi) {
+  Matrix2 X;
   X << 0., -xi.x(),
     xi.x(), 0.;
   return X;
 }
 
 /* ************************************************************************* */
-Rot2::TangentVector Rot2::Vee(const Rot2::LieAlgebra& X) {
+Vector1 Rot2::Vee(const Matrix2& X) {
   TangentVector v;
   v << X(1, 0);
   return v;
diff --git a/gtsam/geometry/Rot2.h b/gtsam/geometry/Rot2.h
index 74a4540cb..85b41e72c 100644
--- a/gtsam/geometry/Rot2.h
+++ b/gtsam/geometry/Rot2.h
@@ -158,10 +158,10 @@ namespace gtsam {
     using LieGroup<Rot2, 1>::inverse; // version with derivative
 
     /// Hat maps from tangent vector to Lie algebra
-    static LieAlgebra Hat(const TangentVector& xi);
+    static Matrix2 Hat(const Vector1& xi);
 
     /// Vee maps from Lie algebra to tangent vector
-    static TangentVector Vee(const LieAlgebra& X);
+    static Vector1 Vee(const Matrix2& X);
 
     /// @}
     /// @name Group Action on Point2
@@ -239,9 +239,9 @@ namespace gtsam {
   };
 
   template<>
-  struct traits<Rot2> : public internal::LieGroup<Rot2> {};
+  struct traits<Rot2> : public internal::MatrixLieGroup<Rot2> {};
 
   template<>
-  struct traits<const Rot2> : public internal::LieGroup<Rot2> {};
+  struct traits<const Rot2> : public internal::MatrixLieGroup<Rot2> {};
 
 } // gtsam
diff --git a/gtsam/geometry/Rot3.h b/gtsam/geometry/Rot3.h
index 09977816c..5768399f8 100644
--- a/gtsam/geometry/Rot3.h
+++ b/gtsam/geometry/Rot3.h
@@ -410,10 +410,10 @@ class GTSAM_EXPORT Rot3 : public LieGroup<Rot3, 3> {
     using LieGroup<Rot3, 3>::inverse; // version with derivative
 
     /// Hat maps from tangent vector to Lie algebra
-    static inline LieAlgebra Hat(const TangentVector& xi) { return SO3::Hat(xi); }
+    static inline Matrix3 Hat(const Vector3& xi) { return SO3::Hat(xi); }
 
     /// Vee maps from Lie algebra to tangent vector
-    static inline TangentVector Vee(const LieAlgebra& X) { return SO3::Vee(X); }
+    static inline Vector3 Vee(const Matrix3& X) { return SO3::Vee(X); }
 
     /// @}
     /// @name Group Action on Point3
@@ -589,10 +589,10 @@ class GTSAM_EXPORT Rot3 : public LieGroup<Rot3, 3> {
       const Matrix3& A, OptionalJacobian<3, 9> H = {});
 
   template<>
-  struct traits<Rot3> : public internal::LieGroup<Rot3> {};
+  struct traits<Rot3> : public internal::MatrixLieGroup<Rot3> {};
 
   template<>
-  struct traits<const Rot3> : public internal::LieGroup<Rot3> {};
+  struct traits<const Rot3> : public internal::MatrixLieGroup<Rot3> {};
   
 }  // namespace gtsam
 
diff --git a/gtsam/geometry/SO3.h b/gtsam/geometry/SO3.h
index 1bdcd82c7..ee51f4d83 100644
--- a/gtsam/geometry/SO3.h
+++ b/gtsam/geometry/SO3.h
@@ -224,13 +224,13 @@ struct GTSAM_EXPORT DexpFunctor : public ExpmapFunctor {
 }  //  namespace so3
 
 /*
- * Define the traits. internal::LieGroup provides both Lie group and Testable
+ * Define the traits. internal::MatrixLieGroup provides both Lie group and Testable
  */
 
 template <>
-struct traits<SO3> : public internal::LieGroup<SO3> {};
+struct traits<SO3> : public internal::MatrixLieGroup<SO3> {};
 
 template <>
-struct traits<const SO3> : public internal::LieGroup<SO3> {};
+struct traits<const SO3> : public internal::MatrixLieGroup<SO3> {};
 
 }  // end namespace gtsam
diff --git a/gtsam/geometry/SO4.h b/gtsam/geometry/SO4.h
index 43b74f7aa..6a37f84bb 100644
--- a/gtsam/geometry/SO4.h
+++ b/gtsam/geometry/SO4.h
@@ -106,13 +106,13 @@ void serialize(Archive &ar, SO4 &Q, const unsigned int /*version*/) {
 #endif
 
 /*
- * Define the traits. internal::LieGroup provides both Lie group and Testable
+ * Define the traits. internal::MatrixLieGroup provides both Lie group and Testable
  */
 
 template <>
-struct traits<SO4> : public internal::LieGroup<SO4> {};
+struct traits<SO4> : public internal::MatrixLieGroup<SO4> {};
 
 template <>
-struct traits<const SO4> : public internal::LieGroup<SO4> {};
+struct traits<const SO4> : public internal::MatrixLieGroup<SO4> {};
 
 }  // end namespace gtsam
diff --git a/gtsam/geometry/SOn.h b/gtsam/geometry/SOn.h
index 910b848fd..b24070c91 100644
--- a/gtsam/geometry/SOn.h
+++ b/gtsam/geometry/SOn.h
@@ -396,14 +396,14 @@ void serialize(
 #endif
 
 /*
- * Define the traits. internal::LieGroup provides both Lie group and Testable
+ * Define the traits. internal::MatrixLieGroup provides both Lie group and Testable
  */
 
 template <int N>
-struct traits<SO<N>> : public internal::LieGroup<SO<N>> {};
+struct traits<SO<N>> : public internal::MatrixLieGroup<SO<N>> {};
 
 template <int N>
-struct traits<const SO<N>> : public internal::LieGroup<SO<N>> {};
+struct traits<const SO<N>> : public internal::MatrixLieGroup<SO<N>> {};
 
 }  // namespace gtsam
 
diff --git a/gtsam/geometry/Similarity2.h b/gtsam/geometry/Similarity2.h
index 9a27842ac..cc4ffc792 100644
--- a/gtsam/geometry/Similarity2.h
+++ b/gtsam/geometry/Similarity2.h
@@ -170,10 +170,10 @@ class GTSAM_EXPORT Similarity2 : public LieGroup<Similarity2, 4> {
   using LieGroup<Similarity2, 4>::inverse;
 
   /// Hat maps from tangent vector to Lie algebra
-  static LieAlgebra Hat(const TangentVector& xi);
+  static Matrix3 Hat(const Vector4& xi);
 
   /// Vee maps from Lie algebra to tangent vector
-  static TangentVector Vee(const LieAlgebra& X);
+  static Vector4 Vee(const Matrix3& X);
 
   /// @}
   /// @name Standard interface
@@ -201,9 +201,9 @@ class GTSAM_EXPORT Similarity2 : public LieGroup<Similarity2, 4> {
 };
 
 template <>
-struct traits<Similarity2> : public internal::LieGroup<Similarity2> {};
+struct traits<Similarity2> : public internal::MatrixLieGroup<Similarity2> {};
 
 template <>
-struct traits<const Similarity2> : public internal::LieGroup<Similarity2> {};
+struct traits<const Similarity2> : public internal::MatrixLieGroup<Similarity2> {};
 
 }  // namespace gtsam
diff --git a/gtsam/geometry/Similarity3.h b/gtsam/geometry/Similarity3.h
index dc66e8721..6eb1d5584 100644
--- a/gtsam/geometry/Similarity3.h
+++ b/gtsam/geometry/Similarity3.h
@@ -174,10 +174,10 @@ class GTSAM_EXPORT Similarity3 : public LieGroup<Similarity3, 7> {
    * @param xi 7-dim twist (w,u,lambda) where
    * @return 4*4 element of Lie algebra that can be exponentiated
    */
-  static LieAlgebra Hat(const TangentVector& xi);
+  static Matrix4 Hat(const Vector7& xi);
 
   /// Vee maps from Lie algebra to tangent vector
-  static TangentVector Vee(const LieAlgebra& X);
+  static Vector7 Vee(const Matrix4& X);
 
   /// @}
   /// @name Standard interface
@@ -243,9 +243,9 @@ inline Matrix wedge<Similarity3>(const Vector& xi) {
 }
 #endif
 template <>
-struct traits<Similarity3> : public internal::LieGroup<Similarity3> {};
+struct traits<Similarity3> : public internal::MatrixLieGroup<Similarity3> {};
 
 template <>
-struct traits<const Similarity3> : public internal::LieGroup<Similarity3> {};
+struct traits<const Similarity3> : public internal::MatrixLieGroup<Similarity3> {};
 
 }  // namespace gtsam
diff --git a/gtsam/geometry/tests/testPose2.cpp b/gtsam/geometry/tests/testPose2.cpp
index 54eb80669..9bbc8f5ca 100644
--- a/gtsam/geometry/tests/testPose2.cpp
+++ b/gtsam/geometry/tests/testPose2.cpp
@@ -37,7 +37,7 @@ GTSAM_CONCEPT_LIE_INST(Pose2)
 TEST(Pose2 , Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<Pose2 >);
   GTSAM_CONCEPT_ASSERT(IsManifold<Pose2 >);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<Pose2 >);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<Pose2 >);
 }
 
 /* ************************************************************************* */
diff --git a/gtsam/geometry/tests/testPose3.cpp b/gtsam/geometry/tests/testPose3.cpp
index a4a6eb574..d14af7ac0 100644
--- a/gtsam/geometry/tests/testPose3.cpp
+++ b/gtsam/geometry/tests/testPose3.cpp
@@ -45,7 +45,7 @@ static const double tol=1e-5;
 TEST(Pose3 , Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<Pose3 >);
   GTSAM_CONCEPT_ASSERT(IsManifold<Pose3 >);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<Pose3 >);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<Pose3 >);
 }
 
 /* ************************************************************************* */
diff --git a/gtsam/geometry/tests/testQuaternion.cpp b/gtsam/geometry/tests/testQuaternion.cpp
index c99c67e0f..40781b832 100644
--- a/gtsam/geometry/tests/testQuaternion.cpp
+++ b/gtsam/geometry/tests/testQuaternion.cpp
@@ -31,7 +31,7 @@ typedef traits<Q>::ChartJacobian QuaternionJacobian;
 TEST(Quaternion , Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<Quaternion >);
   GTSAM_CONCEPT_ASSERT(IsManifold<Quaternion >);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<Quaternion >);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<Quaternion >);
 }
 
 //******************************************************************************
diff --git a/gtsam/geometry/tests/testRot3.cpp b/gtsam/geometry/tests/testRot3.cpp
index 76cd99919..65b97fc09 100644
--- a/gtsam/geometry/tests/testRot3.cpp
+++ b/gtsam/geometry/tests/testRot3.cpp
@@ -40,7 +40,7 @@ static double error = 1e-9, epsilon = 0.001;
 TEST(Rot3 , Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<Rot3 >);
   GTSAM_CONCEPT_ASSERT(IsManifold<Rot3 >);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<Rot3 >);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<Rot3 >);
 }
 
 /* ************************************************************************* */
diff --git a/gtsam/geometry/tests/testSO3.cpp b/gtsam/geometry/tests/testSO3.cpp
index 17b27daea..3f06624e1 100644
--- a/gtsam/geometry/tests/testSO3.cpp
+++ b/gtsam/geometry/tests/testSO3.cpp
@@ -38,7 +38,7 @@ TEST(SO3, Identity) {
 TEST(SO3, Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<SO3>);
   GTSAM_CONCEPT_ASSERT(IsManifold<SO3>);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<SO3>);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<SO3>);
 }
 
 //******************************************************************************
diff --git a/gtsam/geometry/tests/testSO4.cpp b/gtsam/geometry/tests/testSO4.cpp
index 7a3198901..b084ace35 100644
--- a/gtsam/geometry/tests/testSO4.cpp
+++ b/gtsam/geometry/tests/testSO4.cpp
@@ -44,7 +44,7 @@ TEST(SO4, Identity) {
 TEST(SO4, Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<SO4>);
   GTSAM_CONCEPT_ASSERT(IsManifold<SO4>);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<SO4>);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<SO4>);
 }
 
 //******************************************************************************
diff --git a/gtsam/geometry/tests/testSOn.cpp b/gtsam/geometry/tests/testSOn.cpp
index 741ddd3d0..b9609116b 100644
--- a/gtsam/geometry/tests/testSOn.cpp
+++ b/gtsam/geometry/tests/testSOn.cpp
@@ -86,7 +86,7 @@ TEST(SOn, SO5) {
 TEST(SOn, Concept) {
   GTSAM_CONCEPT_ASSERT(IsGroup<SOn>);
   GTSAM_CONCEPT_ASSERT(IsManifold<SOn>);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<SOn>);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<SOn>);
 }
 
 //******************************************************************************
diff --git a/gtsam/geometry/tests/testSimilarity2.cpp b/gtsam/geometry/tests/testSimilarity2.cpp
index b1af74aa8..78b25b50f 100644
--- a/gtsam/geometry/tests/testSimilarity2.cpp
+++ b/gtsam/geometry/tests/testSimilarity2.cpp
@@ -37,7 +37,7 @@ static const double s = 4;
 TEST(Similarity2, Concepts) {
   GTSAM_CONCEPT_ASSERT(IsGroup<Similarity2>);
   GTSAM_CONCEPT_ASSERT(IsManifold<Similarity2>);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<Similarity2>);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<Similarity2>);
 }
 
 //******************************************************************************
diff --git a/gtsam/geometry/tests/testSimilarity3.cpp b/gtsam/geometry/tests/testSimilarity3.cpp
index 86a4ad940..ce0814f7c 100644
--- a/gtsam/geometry/tests/testSimilarity3.cpp
+++ b/gtsam/geometry/tests/testSimilarity3.cpp
@@ -56,7 +56,7 @@ const double degree = M_PI / 180;
 TEST(Similarity3, Concepts) {
   GTSAM_CONCEPT_ASSERT(IsGroup<Similarity3 >);
   GTSAM_CONCEPT_ASSERT(IsManifold<Similarity3 >);
-  GTSAM_CONCEPT_ASSERT(IsLieGroup<Similarity3 >);
+  GTSAM_CONCEPT_ASSERT(IsMatrixLieGroup<Similarity3 >);
 }
 
 //******************************************************************************