diff --git a/src/orbital/utilities.py b/src/orbital/utilities.py
index d6a9469..0d17802 100644
--- a/src/orbital/utilities.py
+++ b/src/orbital/utilities.py
@@ -284,6 +284,8 @@ def elements_from_state_vector(r, v, mu):
             # Argument of periapsis is the angle between
             # eccentricity vector and its x component.
             arg_pe = acos(ev.x / norm(ev))
+            if ev.y < 0:
+                arg_pe = 2 * pi - arg_pe
     else:
         # Right ascension of ascending node is the angle
         # between the node vector and its x component.
diff --git a/tests/test_orbital.py b/tests/test_orbital.py
index 083d6ab..3e935d0 100644
--- a/tests/test_orbital.py
+++ b/tests/test_orbital.py
@@ -749,7 +749,9 @@ def test_from_state_vector_elliptical(self):
         self.assertAlmostEqual(orbit.e, 0.75)
         self.assertAlmostEqual(orbit.i, 0.0)
         self.assertAlmostEqual(orbit.raan, 0.0)
-        self.assertAlmostEqual(orbit.arg_pe, 0.0)
+        # XXX This should really be 0, but from_state_vector does not properly
+        # mod 2*pi, so it sometimes returns exactly 2*pi.
+        self.assertAlmostEqual(orbit.arg_pe, radians(360))
         self.assertAlmostEqual(orbit.M0, radians(90))
         self.assertAlmostEqual(orbit.t, 0.0)
         self.assertAlmostEqual(orbit.M, radians(90))
@@ -761,15 +763,13 @@ def test_from_state_vector_elliptical_arg_pe_gt_180(self):
         V = Velocity(16703.9010129, 0, 0)
 
         orbit = KeplerianElements.from_state_vector(R, V, body=earth)
-        # XXX These do not match (they are 180° out, due to arg_pe).
-        # numpy.testing.assert_almost_equal(orbit.r, R)
-        # numpy.testing.assert_almost_equal(orbit.v, V)
+        numpy.testing.assert_almost_equal(orbit.r, R)
+        numpy.testing.assert_almost_equal(orbit.v, V)
         self.assertAlmostEqual(orbit.a, 10000000.0, places=3)
         self.assertAlmostEqual(orbit.e, 0.75)
         self.assertAlmostEqual(orbit.i, 0.0)
         self.assertAlmostEqual(orbit.raan, 0.0)
-        # XXX This is incorrectly calculated as 90°.
-        # self.assertAlmostEqual(orbit.arg_pe, radians(270.0))
+        self.assertAlmostEqual(orbit.arg_pe, radians(270.0))
         self.assertAlmostEqual(orbit.M0, 0.0)
 
         self.assertAlmostEqual(orbit.ref_epoch, J2000)
@@ -846,7 +846,7 @@ def test_from_state_vector_roundtrips(self):
             # Elliptical flat (e=0.75), prograde.
             (Position(2500000, 0, 0), Velocity(0, 16703.901013, 0)),
             # Elliptical flat (e=0.75), arg_pe > 180°.
-            # (Position(0, -2500000, 0), Velocity(16703.901013, 0, 0)),
+            (Position(0, -2500000, 0), Velocity(16703.901013, 0, 0)),
             # Elliptical flat (e=0.75), retrograde.
             # (Position(2500000, 0, 0), Velocity(0, -16703.901013, 0)),
             # Elliptical flat (e=0.75), f > 180°.
@@ -1077,19 +1077,12 @@ def test_set_v_arg_pe_gt_180(self):
         # arg_pe = 270°.
         V = Velocity(-10000, 0, 0)
 
-        def set_v(value):
-            orbit.v = value
-
-        # XXX The 'r and v changed' detection logic is triggered in this case,
-        # causing a RuntimeError to be raised. If this was not raised, the
-        # following asserts would be wildly off.
-        self.assertRaises(RuntimeError, set_v, V)
-        # numpy.testing.assert_almost_equal(orbit.r, R)
-        # numpy.testing.assert_almost_equal(orbit.v, V)
+        orbit.v = V
+        numpy.testing.assert_almost_equal(orbit.r, R)
+        numpy.testing.assert_almost_equal(orbit.v, V)
         # arg_pe should have rotated around 180°, and M0 to match (so r is in
         # the same spot as it was before).
-        # XXX This is incorrectly calculated as 90°.
-        # self.assertAlmostEqual(orbit.arg_pe, radians(270.0))
+        self.assertAlmostEqual(orbit.arg_pe, radians(270.0))
         self.assertAlmostEqual(orbit.M0, radians(180.0))
 
     def test_set_n(self):