diff --git a/popcor/examples/rotation_lifter.py b/popcor/examples/rotation_lifter.py
index 6a7cf97..7d824ed 100644
--- a/popcor/examples/rotation_lifter.py
+++ b/popcor/examples/rotation_lifter.py
@@ -1,5 +1,6 @@
 """RotationAveraging class for rotation averaging and synchronization problems."""
 
+import warnings
 from typing import Any, Dict, List, Tuple
 
 import numpy as np
@@ -8,6 +9,7 @@
 from scipy.spatial.transform import Rotation
 
 from popcor.base_lifters import StateLifter
+from popcor.utils.plotting_tools import LINESTYLES
 
 METHOD: str = "CG"
 SOLVER_KWARGS: dict = dict(
@@ -174,8 +176,16 @@ def get_theta(self, x: np.ndarray) -> np.ndarray:
             return C_flat.reshape((self.d, self.n_rot * self.d), order="F")
         elif self.level == "bm":
             R0 = x[: self.d, : self.d].T
+
+            I = R0.T @ R0
+            scale_R = I[0, 0]
+            np.testing.assert_allclose(np.diag(I), scale_R)
+            warnings.warn(f"R0 is scaled by {scale_R}", UserWarning)
+
             Ri = np.array(x[self.d : (self.n_rot + 1) * self.d, : self.d]).T
-            Ri_world = R0.T @ Ri
+            np.testing.assert_allclose(np.diag(Ri.T @ Ri), scale_R)
+            Ri_world = R0.T @ Ri / scale_R
+            np.testing.assert_allclose(np.diag(Ri_world.T @ Ri_world), 1.0)
             return Ri_world
         else:
             raise ValueError(f"Unknown level {self.level} for RotationLifter")
@@ -448,14 +458,15 @@ def plot(self, estimates: Dict[str, np.ndarray] = {}) -> Tuple[Any, Any]:
             )
             label = None
 
-        linestyles = itertools.cycle(["--", "-.", ":"])
+        linestyles = itertools.cycle(LINESTYLES)
         for label, theta in estimates.items():
+            ls = next(linestyles)
             for i in range(self.n_rot):
                 plot_frame(
                     ax=ax,
                     theta=theta[:, i * self.d : (i + 1) * self.d],
                     label=label,
-                    ls=next(linestyles),
+                    ls=ls,
                     scale=1.0,
                     marker="",
                     r_wc_w=np.hstack([i * 2.0] + [0.0] * (self.d - 1)),  # type: ignore
diff --git a/popcor/utils/plotting_tools.py b/popcor/utils/plotting_tools.py
index 94f1633..bdf129b 100644
--- a/popcor/utils/plotting_tools.py
+++ b/popcor/utils/plotting_tools.py
@@ -6,6 +6,19 @@
 
 from popcor.utils.geometry import get_C_r_from_theta
 
+LINESTYLES = [
+    "-",  # solid
+    "--",  # dashed
+    "-.",  # dash-dot
+    ":",  # dotted
+    (0, (5, 2)),  # long dash, short gap
+    (0, (1, 1)),  # very fine dashed (dot style)
+    (0, (3, 5, 1, 5)),  # dash, gap, dot, gap
+    (0, (5, 10)),  # widely spaced dashes
+    (0, (3, 1, 1, 1, 1, 1)),  # dash-dot-dot
+    (0, (2, 4, 6, 4)),  # mixed dash lengths
+]
+
 
 def import_plt():
     import shutil
diff --git a/tests/test_rotation_lifter.py b/tests/test_rotation_lifter.py
index 42668c7..931225d 100644
--- a/tests/test_rotation_lifter.py
+++ b/tests/test_rotation_lifter.py
@@ -135,16 +135,17 @@ def test_solve_local():
     print("done")
 
 
-def test_solve_sdp():
+@pytest.mark.parametrize("level", ["no", "bm"])
+def test_solve_sdp(level):
     """Solve the SDP relaxation and compare the recovered rotations to ground truth."""
     d = 3
     n_abs = 1
-    n_rot = 4
+    n_rot = 5
     estimates = {}
-    for noise in [0.0, 0.2]:
+    for noise in [0, 1e-3, 0.2]:
         np.random.seed(2)
         lifter = RotationLifter(
-            d=d, n_abs=n_abs, n_rot=n_rot, sparsity="chain", level="no"
+            d=d, n_abs=n_abs, n_rot=n_rot, sparsity="chain", level=level
         )
 
         y = lifter.simulate_y(noise=noise)
@@ -160,13 +161,23 @@ def test_solve_sdp():
 
         X, info = solve_sdp(Q, constraints, verbose=False)
 
-        x, info_rank = rank_project(X, p=1)
+        if lifter.level == "no":
+            x, info_rank = rank_project(X, p=1)
+            theta_sdp = lifter.get_theta(x)
+        else:
+            X, info_rank = rank_project(X, p=lifter.d)
+            theta_sdp = lifter.get_theta(X)
+
         print(f"EVR at noise {noise:.2f}: {info_rank['EVR']:.2e}")
         if noise <= 1:
             assert info_rank["EVR"] > 1e8
-        theta_sdp = lifter.get_theta(x)
+
+        error = lifter.get_error(theta_sdp)
         if noise == 0.0:
-            np.testing.assert_allclose(theta_sdp, lifter.theta, atol=1e-5)
+            np.testing.assert_allclose(theta_sdp, lifter.theta, atol=1e-8)
+            assert error < 1e-10
+        elif noise < 1e-2:
+            assert error < 1e-2
 
         estimates.update({"init gt": lifter.theta, f"SDP noise {noise:.2f}": theta_sdp})
 
@@ -178,7 +189,8 @@ def test_solve_sdp():
 
 
 if __name__ == "__main__":
-    test_solve_sdp()
+    test_solve_sdp(level="no")
+    test_solve_sdp(level="bm")
     test_solve_local()
 
     test_measurements(d=2, level="no")