diff --git a/src/sage/categories/coxeter_groups.py b/src/sage/categories/coxeter_groups.py
index 3579487bcb1..94d7824949f 100644
--- a/src/sage/categories/coxeter_groups.py
+++ b/src/sage/categories/coxeter_groups.py
@@ -259,7 +259,7 @@ def coxeter_element(self):
                 element.
 
                 In this context, this is an element having a regular
-                eigenvector (a vector not contained in any reflecting
+                eigenvector (a vector not contained in any reflection
                 hyperplane of ``self``).
 
             EXAMPLES::
diff --git a/src/sage/categories/finite_complex_reflection_groups.py b/src/sage/categories/finite_complex_reflection_groups.py
index 82b80c9d5da..8416f75a926 100644
--- a/src/sage/categories/finite_complex_reflection_groups.py
+++ b/src/sage/categories/finite_complex_reflection_groups.py
@@ -195,10 +195,11 @@ def _test_degrees(self, **options):
                 Reducible real reflection group of rank 4 and type A2 x B2
                 sage: W._test_degrees()
 
-                sage: SymmetricGroup(3)._test_degrees()
+                sage: W = SymmetricGroup(5)
+                sage: W._test_degrees()
 
+            We now break the implementation of W.degrees and check that this is caught::
 
-                sage: W = SymmetricGroup(5)
                 sage: W.degrees = lambda: (1/1,5)
                 sage: W._test_degrees()
                 Traceback (most recent call last):
@@ -211,6 +212,11 @@ def _test_degrees(self, **options):
                 ...
                 AssertionError: the degrees should be larger than 2
 
+            We restore W to its normal state::
+
+                sage: del W.degrees
+                sage: W._test_degrees()
+
             See the documentation for :class:`TestSuite` for more information.
             """
             from sage.structure.element import parent
@@ -244,9 +250,11 @@ def _test_codegrees(self, **options):
                 Reducible real reflection group of rank 4 and type A2 x B2
                 sage: W._test_codegrees()
 
-                sage: SymmetricGroup(3)._test_codegrees()
-
                 sage: W = SymmetricGroup(5)
+                sage: W._test_codegrees()
+
+            We now break the implementation of W.degrees and check that this is caught::
+
                 sage: W.codegrees = lambda: (1/1,5)
                 sage: W._test_codegrees()
                 Traceback (most recent call last):
@@ -259,6 +267,11 @@ def _test_codegrees(self, **options):
                 ...
                 AssertionError: the codegrees should be nonnegative
 
+            We restore W to its normal state::
+
+                sage: del W.codegrees
+                sage: W._test_codegrees()
+
             See the documentation for :class:`TestSuite` for more information.
             """
             from sage.structure.element import parent
@@ -279,7 +292,7 @@ def _test_codegrees(self, **options):
 
         def number_of_reflection_hyperplanes(self):
             r"""
-            Return the number of reflecting hyperplanes of ``self``.
+            Return the number of reflection hyperplanes of ``self``.
 
             This is also the number of distinguished reflections.  For
             real groups, this coincides with the number of
@@ -306,14 +319,15 @@ def number_of_reflection_hyperplanes(self):
                 sage: W.number_of_reflection_hyperplanes()
                 15
             """
-            return sum(self.codegrees()) + self.rank()
+            from sage.rings.all import ZZ
+            return ZZ.sum(self.codegrees()) + self.rank()
 
         def number_of_reflections(self):
             r"""
             Return the number of reflections of ``self``.
 
             For real groups, this coincides with the number of
-            reflecting hyperplanes.
+            reflection hyperplanes.
 
             This implementation uses that it is given by the sum of
             the degrees of ``self`` minus its rank.
@@ -571,7 +585,7 @@ def coxeter_number(self):
 
                 This is defined as `\frac{N + N^*}{n}` where
                 `N` is the number of reflections, `N^*` is the
-                number of reflecting hyperplanes, and `n` is the
+                number of reflection hyperplanes, and `n` is the
                 rank of ``self``.
 
                 EXAMPLES::
diff --git a/src/sage/combinat/root_system/reflection_group_complex.py b/src/sage/combinat/root_system/reflection_group_complex.py
index 5f1bf63d016..db121290c75 100644
--- a/src/sage/combinat/root_system/reflection_group_complex.py
+++ b/src/sage/combinat/root_system/reflection_group_complex.py
@@ -539,7 +539,7 @@ def distinguished_reflection(self, i):
     @cached_method
     def reflection_hyperplanes(self, as_linear_functionals=False):
         r"""
-        Return the list of all reflecting hyperplanes of ``self``,
+        Return the list of all reflection hyperplanes of ``self``,
         either as a codimension 1 space, or as its linear functional.
 
         INPUT:
@@ -601,9 +601,9 @@ def reflection_hyperplanes(self, as_linear_functionals=False):
 
     def reflection_hyperplane(self, i, as_linear_functional=False):
         r"""
-        Return the ``i``-th reflecting hyperplane of ``self``.
+        Return the ``i``-th reflection hyperplane of ``self``.
 
-        The ``i``-th reflecting hyperplane corresponds to the ``i``
+        The ``i``-th reflection hyperplane corresponds to the ``i``
         distinguished reflection.
 
         INPUT:
@@ -2280,7 +2280,7 @@ def is_h_regular(self, is_class_representative=False):
             Return whether ``self`` is regular.
 
             This is if ``self`` has an eigenvector with eigenvalue `h`
-            and which does not lie in any reflecting hyperplane.
+            and which does not lie in any reflection hyperplane.
             Here, `h` denotes the Coxeter number.
 
             EXAMPLES::
@@ -2308,7 +2308,7 @@ def is_regular(self, h, is_class_representative=False):
             Return whether ``self`` is regular.
 
             This is, if ``self`` has an eigenvector with eigenvalue
-            ``h`` and which does not lie in any reflecting hyperplane.
+            ``h`` and which does not lie in any reflection hyperplane.
 
             - ``is_class_representative`` -- boolean (default ``True``) whether to
               compute instead on the conjugacy class representative.