diff --git a/.github/scripts/unix.sh b/.github/scripts/unix.sh
index f85e67dc1..55a8ac372 100644
--- a/.github/scripts/unix.sh
+++ b/.github/scripts/unix.sh
@@ -66,6 +66,8 @@ function configure()
       -DGTSAM_BUILD_EXAMPLES_ALWAYS=${GTSAM_BUILD_EXAMPLES_ALWAYS:-ON} \
       -DGTSAM_ALLOW_DEPRECATED_SINCE_V41=${GTSAM_ALLOW_DEPRECATED_SINCE_V41:-OFF} \
       -DGTSAM_USE_QUATERNIONS=${GTSAM_USE_QUATERNIONS:-OFF} \
+      -DGTSAM_ROT3_EXPMAP=${GTSAM_ROT3_EXPMAP:-ON} \
+      -DGTSAM_POSE3_EXPMAP=${GTSAM_POSE3_EXPMAP:-ON} \
       -DGTSAM_BUILD_WITH_MARCH_NATIVE=OFF \
       -DBOOST_ROOT=$BOOST_ROOT \
       -DBoost_NO_SYSTEM_PATHS=ON \
diff --git a/.github/workflows/build-special.yml b/.github/workflows/build-special.yml
index 715f25733..532f917c1 100644
--- a/.github/workflows/build-special.yml
+++ b/.github/workflows/build-special.yml
@@ -24,6 +24,7 @@ jobs:
             ubuntu-gcc-deprecated,
             ubuntu-gcc-quaternions,
             ubuntu-gcc-tbb,
+            ubuntu-cayleymap,
           ]
 
         build_type: [Debug, Release]
@@ -47,6 +48,12 @@ jobs:
             version: "9"
             flag: tbb
 
+          - name: ubuntu-cayleymap
+            os: ubuntu-18.04
+            compiler: gcc
+            version: "9"
+            flag: cayley
+
     steps:
       - name: Checkout
         uses: actions/checkout@master
@@ -109,6 +116,13 @@ jobs:
           echo "GTSAM_WITH_TBB=ON" >> $GITHUB_ENV
           echo "GTSAM Uses TBB"
 
+      - name: Use Cayley Transform for Rot3
+        if: matrix.flag == 'cayley'
+        run: |
+          echo "GTSAM_POSE3_EXPMAP=OFF" >> $GITHUB_ENV
+          echo "GTSAM_ROT3_EXPMAP=OFF" >> $GITHUB_ENV
+          echo "GTSAM Uses Cayley map for Rot3"
+
       - name: Build & Test
         run: |
           bash .github/scripts/unix.sh -t
diff --git a/cmake/HandleGeneralOptions.cmake b/cmake/HandleGeneralOptions.cmake
index 27d73bd86..85a529e49 100644
--- a/cmake/HandleGeneralOptions.cmake
+++ b/cmake/HandleGeneralOptions.cmake
@@ -31,6 +31,15 @@ if(NOT MSVC AND NOT XCODE_VERSION)
     option(GTSAM_BUILD_WITH_CCACHE           "Use ccache compiler cache" ON)
 endif()
 
+# Enable GTSAM_ROT3_EXPMAP if GTSAM_POSE3_EXPMAP is enabled, and vice versa.
+if(GTSAM_POSE3_EXPMAP)
+    message(STATUS "GTSAM_POSE3_EXPMAP=ON, enabling GTSAM_ROT3_EXPMAP as well")
+    set(GTSAM_ROT3_EXPMAP 1 CACHE BOOL "" FORCE)
+elseif(GTSAM_ROT3_EXPMAP)
+    message(STATUS "GTSAM_ROT3_EXPMAP=ON, enabling GTSAM_POSE3_EXPMAP as well")
+    set(GTSAM_POSE3_EXPMAP 1 CACHE BOOL "" FORCE)
+endif()
+
 # Options relating to MATLAB wrapper
 # TODO: Check for matlab mex binary before handling building of binaries
 option(GTSAM_INSTALL_MATLAB_TOOLBOX      "Enable/Disable installation of matlab toolbox"  OFF)
diff --git a/gtsam/geometry/Rot3M.cpp b/gtsam/geometry/Rot3M.cpp
index 02e5b771f..07cc7302a 100644
--- a/gtsam/geometry/Rot3M.cpp
+++ b/gtsam/geometry/Rot3M.cpp
@@ -176,7 +176,17 @@ Vector3 Rot3::CayleyChart::Local(const Rot3& R, OptionalJacobian<3,3> H) {
   if (H) throw std::runtime_error("Rot3::CayleyChart::Local Derivative");
   // Create a fixed-size matrix
   Matrix3 A = R.matrix();
-  // Mathematica closed form optimization (procrastination?) gone wild:
+
+  // Check if (A+I) is invertible. Same as checking for -1 eigenvalue.
+  if ((A + I_3x3).determinant() == 0.0) {
+    throw std::runtime_error("Rot3::CayleyChart::Local Invalid Rotation");
+  }
+
+  // Mathematica closed form optimization.
+  // The following are the essential computations for the following algorithm
+  // 1. Compute the inverse of P = (A+I), using a closed-form formula since P is 3x3 
+  // 2. Compute the Cayley transform C = 2 * P^{-1} * (A-I)
+  // 3. C is skew-symmetric, so we pick out the computations corresponding only to x, y, and z.
   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);
diff --git a/gtsam/geometry/tests/testPose3.cpp b/gtsam/geometry/tests/testPose3.cpp
index 9639ffbcf..8fafcb25f 100644
--- a/gtsam/geometry/tests/testPose3.cpp
+++ b/gtsam/geometry/tests/testPose3.cpp
@@ -906,9 +906,9 @@ TEST(Pose3 , ChartDerivatives) {
   Pose3 id;
   if (ROT3_DEFAULT_COORDINATES_MODE == Rot3::EXPMAP) {
     CHECK_CHART_DERIVATIVES(id,id);
-//    CHECK_CHART_DERIVATIVES(id,T2);
-//    CHECK_CHART_DERIVATIVES(T2,id);
-//    CHECK_CHART_DERIVATIVES(T2,T3);
+    CHECK_CHART_DERIVATIVES(id,T2);
+    CHECK_CHART_DERIVATIVES(T2,id);
+    CHECK_CHART_DERIVATIVES(T2,T3);
   }
 }
 
diff --git a/gtsam_unstable/geometry/tests/testSimilarity3.cpp b/gtsam_unstable/geometry/tests/testSimilarity3.cpp
index b985eb374..937761f02 100644
--- a/gtsam_unstable/geometry/tests/testSimilarity3.cpp
+++ b/gtsam_unstable/geometry/tests/testSimilarity3.cpp
@@ -361,8 +361,12 @@ TEST(Similarity3, AlignPose3) {
 
   vector<Pose3Pair> correspondences{bTa1, bTa2};
 
+  // Cayley transform cannot accommodate 180 degree rotations,
+  // hence we only test for Expmap
+#ifdef GTSAM_ROT3_EXPMAP
   Similarity3 actual_aSb = Similarity3::Align(correspondences);
   EXPECT(assert_equal(expected_aSb, actual_aSb));
+#endif
 }
 
 //******************************************************************************