diff --git a/gtsam/geometry/Rot3M.cpp b/gtsam/geometry/Rot3M.cpp
index 77d97219c..e0d3086e9 100644
--- a/gtsam/geometry/Rot3M.cpp
+++ b/gtsam/geometry/Rot3M.cpp
@@ -270,23 +270,23 @@ Vector3 Rot3::localCoordinates(const Rot3& T, Rot3::CoordinatesMode mode) const
     return Logmap(between(T));
   } else if(mode == Rot3::CAYLEY) {
     // Create a fixed-size matrix
-    Eigen::Matrix3d A(between(T).matrix());
+    Matrix3 A = rot_.transpose() * T.matrix();
     // Mathematica closed form optimization (procrastination?) gone wild:
     const double a=A(0,0),b=A(0,1),c=A(0,2);
     const double d=A(1,0),e=A(1,1),f=A(1,2);
     const double g=A(2,0),h=A(2,1),i=A(2,2);
     const double di = d*i, ce = c*e, cd = c*d, fg=f*g;
     const double M = 1 + e - f*h + i + e*i;
-    const double K = 2.0 / (cd*h + M + a*M -g*(c + ce) - b*(d + di - fg));
-    const double x = (a * f - cd + f) * K;
-    const double y = (b * f - ce - c) * K;
-    const double z = (fg - di - d) * K;
-    return -2 * Vector3(x, y, z);
+    const double K = - 4.0 / (cd*h + M + a*M -g*(c + ce) - b*(d + di - fg));
+    const double x = a * f - cd + f;
+    const double y = b * f - ce - c;
+    const double z = fg - di - d;
+    return K * Vector3(x, y, z);
   } else if(mode == Rot3::SLOW_CAYLEY) {
     // Create a fixed-size matrix
-    Eigen::Matrix3d A(between(T).matrix());
+    Matrix3 A(between(T).matrix());
     // using templated version of Cayley
-    Eigen::Matrix3d Omega = CayleyFixed<3>(A);
+    Matrix3 Omega = CayleyFixed<3>(A);
     return -2*Vector3(Omega(2,1),Omega(0,2),Omega(1,0));
   } else {
     assert(false);
diff --git a/gtsam/geometry/tests/testRot3M.cpp b/gtsam/geometry/tests/testRot3M.cpp
index 08f5a42e9..24a092fbe 100644
--- a/gtsam/geometry/tests/testRot3M.cpp
+++ b/gtsam/geometry/tests/testRot3M.cpp
@@ -145,10 +145,15 @@ TEST( Rot3, rodriguez4)
 }
 
 /* ************************************************************************* */
-TEST( Rot3, expmap)
+TEST( Rot3, retract)
 {
   Vector v = zero(3);
-  CHECK(assert_equal(R.retract(v), R));
+  CHECK(assert_equal(R, R.retract(v)));
+
+  // test Canonical coordinates
+  Canonical<Rot3> chart;
+  Vector v2 = chart.apply(R);
+  CHECK(assert_equal(R, chart.retract(v2)));
 }
 
 /* ************************************************************************* */