Doing reviewer suggestions.

src/sage/algebras/lie_algebras/bgg_dual_module.py

@@ -106,11 +106,11 @@ def __init__(self, module):
 
             sage: g = LieAlgebra(QQ, cartan_type=['B', 2])
             sage: La = g.cartan_type().root_system().weight_space().fundamental_weights()
-            sage: M = g.verma_module(2*La[1]+La[2])
+            sage: M = g.verma_module(2*La[1] + La[2])
             sage: Mc = M.dual()
             sage: TestSuite(Mc).run()
 
-            sage: M = g.verma_module(2/3*La[1]-3/5*La[2])
+            sage: M = g.verma_module(2/3*La[1] - 3/5*La[2])
             sage: Mc = M.dual()
             sage: TestSuite(Mc).run()
         """
@@ -449,7 +449,7 @@ def __classcall__(cls, simple, prefix='f', **kwds):
 
             sage: g = LieAlgebra(QQ, cartan_type=['E', 6])
             sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-            sage: L = g.simple_module(La[1]+La[3])
+            sage: L = g.simple_module(La[1] + La[3])
             sage: from sage.algebras.lie_algebras.bgg_dual_module import SimpleModuleIndices
             sage: SimpleModuleIndices(L) is L._indices
             True
@@ -464,10 +464,10 @@ def __init__(self, simple, prefix, category=None, **kwds):
 
             sage: g = LieAlgebra(QQ, cartan_type=['A', 2])
             sage: La = g.cartan_type().root_system().weight_space().fundamental_weights()
-            sage: I = g.simple_module(2*La[1]+La[2]).indices()
+            sage: I = g.simple_module(2*La[1] + La[2]).indices()
             sage: TestSuite(I).run()
 
-            sage: I = g.simple_module(2*La[1]-1/3*La[2]).indices()
+            sage: I = g.simple_module(2*La[1] - 1/3*La[2]).indices()
             sage: TestSuite(I).run(max_runs=150)  # long time
         """
         self._simple = simple
@@ -509,7 +509,7 @@ def _an_element_(self):
 
             sage: g = LieAlgebra(QQ, cartan_type=['E', 6])
             sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-            sage: I = g.simple_module(2*La[1]+La[2]).indices()
+            sage: I = g.simple_module(2*La[1] + La[2]).indices()
             sage: I._an_element_()
             1
         """
@@ -679,7 +679,7 @@ def _construct_next_level(self):
         .. TODO::
 
             Avoid unnecessary computations by using the corresponding
-            (combinastorial) crystal.
+            (combinatorial) crystal.
 
         EXAMPLES::
 
@@ -800,11 +800,11 @@ def __classcall_private__(cls, g, weight, *args, **kwds):
 
             sage: g = LieAlgebra(QQ, cartan_type=['E', 6])
             sage: La = g.cartan_type().root_system().weight_space().fundamental_weights()
-            sage: type(g.simple_module(La[1]+La[2]))
+            sage: type(g.simple_module(La[1] + La[2]))
             <class 'sage.algebras.lie_algebras.bgg_dual_module.FiniteDimensionalSimpleModule_with_category'>
-            sage: type(g.simple_module(La[1]-La[2]))
+            sage: type(g.simple_module(La[1] - La[2]))
             <class 'sage.algebras.lie_algebras.bgg_dual_module.SimpleModule_with_category'>
-            sage: type(g.simple_module(La[1]+3/2*La[2]))
+            sage: type(g.simple_module(La[1] + 3/2*La[2]))
             <class 'sage.algebras.lie_algebras.bgg_dual_module.SimpleModule_with_category'>
         """
         if weight.is_dominant_weight():
@@ -819,12 +819,12 @@ def __init__(self, g, weight, prefix='f', basis_key=None, **kwds):
 
             sage: g = LieAlgebra(QQ, cartan_type=['G', 2])
             sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-            sage: L = g.simple_module(La[1]+La[2])
+            sage: L = g.simple_module(La[1] + La[2])
             sage: TestSuite(L).run()
 
             sage: g = LieAlgebra(QQ, cartan_type=['A', 2])
             sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-            sage: L = g.simple_module(La[1]-La[2])
+            sage: L = g.simple_module(La[1] - La[2])
             sage: TestSuite(L).run()
         """
         self._g = g
@@ -988,7 +988,7 @@ def dual(self):
 
             sage: g = LieAlgebra(QQ, cartan_type=['B', 4])
             sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-            sage: L = g.simple_module(2*La[1]+3*La[4])
+            sage: L = g.simple_module(2*La[1] + 3*La[4])
             sage: L.dual() is L
             True
         """
@@ -1064,7 +1064,7 @@ def homogeneous_component_basis(self, mu):
             sage: g = LieAlgebra(QQ, cartan_type=['A', 2])
             sage: P = g.cartan_type().root_system().weight_lattice()
             sage: La = P.fundamental_weights()
-            sage: la = La[1]+La[2]
+            sage: la = La[1] + La[2]
             sage: L = g.simple_module(la)
             sage: from itertools import product
             sage: al = P.simple_roots()
@@ -1118,7 +1118,7 @@ def _acted_upon_(self, scalar, self_on_left=True):
 
                 sage: g = LieAlgebra(QQ, cartan_type=['A', 2])
                 sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-                sage: L = g.simple_module(La[1]+La[2])
+                sage: L = g.simple_module(La[1] + La[2])
                 sage: v = L.highest_weight_vector(); v
                 u[Lambda[1] + Lambda[2]]
                 sage: f1, f2 = g.pbw_basis().f()
@@ -1165,7 +1165,7 @@ def bgg_resolution(self):
 
             sage: g = LieAlgebra(QQ, cartan_type=['A', 2])
             sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-            sage: L = g.simple_module(La[1]+La[2])
+            sage: L = g.simple_module(La[1] + La[2])
             sage: L.bgg_resolution()
             BGG resolution of Simple module with highest weight Lambda[1] + Lambda[2]
              of Lie algebra of ['A', 2] in the Chevalley basis

src/sage/algebras/lie_algebras/bgg_resolution.py

@@ -48,7 +48,7 @@ class BGGResolution(UniqueRepresentation, ChainComplex_class):
 
         sage: g = LieAlgebra(QQ, cartan_type=['A', 2])
         sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-        sage: L = g.simple_module(La[1]+4*La[2])
+        sage: L = g.simple_module(La[1] + 4*La[2])
         sage: res = L.bgg_resolution()
         sage: ascii_art(res)
                                 [ 1 -1]       [1]
@@ -57,7 +57,7 @@ class BGGResolution(UniqueRepresentation, ChainComplex_class):
 
         sage: g = LieAlgebra(QQ, cartan_type=['D', 4])
         sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-        sage: L = g.simple_module(La[1]+La[2]+3*La[3])
+        sage: L = g.simple_module(La[1] + La[2] + 3*La[3])
         sage: res = L.bgg_resolution()
         sage: w0 = WeylGroup(g.cartan_type(), prefix='s').long_element()
         sage: all(res.differential(i) * res.differential(i+1) == 0
@@ -72,7 +72,7 @@ def __init__(self, L):
 
             sage: g = LieAlgebra(QQ, cartan_type=['B', 2])
             sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-            sage: L = g.simple_module(La[1]+La[2])
+            sage: L = g.simple_module(La[1] + La[2])
             sage: res = L.bgg_resolution()
             sage: TestSuite(res).run()
         """
@@ -113,7 +113,7 @@ def simple_module(self):
 
             sage: g = LieAlgebra(QQ, cartan_type=['C', 2])
             sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-            sage: L = g.simple_module(La[1]+La[2])
+            sage: L = g.simple_module(La[1] + La[2])
             sage: res = L.bgg_resolution()
             sage: res.simple_module() is L
             True
@@ -226,4 +226,4 @@ def build_differentials(W):
         prev_mat = mat
     differentials[0] = matrix.zero(ZZ, 0, 1)
     differentials[maxlen+1] = matrix.zero(ZZ, 1, 0)
-    return differentials, module_order
+    return differentials, module_order

src/sage/algebras/lie_algebras/poincare_birkhoff_witt.py

@@ -657,7 +657,7 @@ def __init__(self, g, basis_key=None, *args, **kwds):
             sage: PBW = L.pbw_basis()
             sage: E, F, H = PBW.algebra_generators()
             sage: TestSuite(PBW).run(elements=[E, F, H])
-            sage: TestSuite(PBW).run(elements=[E, F, H, E*F + H]) # long time
+            sage: TestSuite(PBW).run(elements=[E, F, H, E*F + H])  # long time
         """
         super().__init__(g, basis_key, *args, **kwds)
         if self._basis_key == self._g._triangular_key:
@@ -801,5 +801,14 @@ def transpose(self):
             EXAMPLES::
 
                 sage: g = LieAlgebra(QQ, cartan_type=['D', 4])
+                sage: U = g.pbw_basis()
+                sage: e = U.e()
+                sage: f = U.f()
+                sage: elts = [e[1], e[1]*e[2], e[3]+e[4], e[1]*e[3]*e[4] + e[2],
+                ....:         f[1], f[1]*f[2], f[3]+f[4], e[1]*e[3]*e[4] + e[2],
+                ....:         e[1]*f[1], f[1]*e[1], (e[2]*f[2] - f[2]*e[2])^2]
+                sage: all((b*bp).transpose() == bp.transpose() * b.transpose()
+                ....:     for b in elts for bp in elts)
+                True
             """
             return self.parent().transpose(self)

src/sage/algebras/lie_algebras/verma_module.py

@@ -35,7 +35,7 @@
 from sage.rings.rational_field import QQ
 
 
-class ModulePrinting(object):
+class ModulePrinting:
     """
     Helper mixin class for printing the module vectors.
     """
@@ -1158,8 +1158,8 @@ def image(self):
 
             sage: g = LieAlgebra(QQ, cartan_type=['B', 2])
             sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-            sage: M = g.verma_module(La[1]+2*La[2])
-            sage: Mp = g.verma_module(La[1]+3*La[2])
+            sage: M = g.verma_module(La[1] + 2*La[2])
+            sage: Mp = g.verma_module(La[1] + 3*La[2])
             sage: phi = Hom(M, Mp).natural_map()
             sage: phi.image()
             Free module generated by {} over Rational Field
@@ -1302,7 +1302,7 @@ def highest_weight_image(self):
 
             sage: g = LieAlgebra(QQ, cartan_type=['C', 3])
             sage: La = g.cartan_type().root_system().weight_lattice().fundamental_weights()
-            sage: M = g.verma_module(La[1]+2*La[3])
+            sage: M = g.verma_module(La[1] + 2*La[3])
             sage: Mc = M.dual()
             sage: H = Hom(M, Mc)
             sage: H.highest_weight_image()

src/sage/categories/triangular_kac_moody_algebras.py

@@ -422,7 +422,7 @@ def transpose(self):
                 The transpose `\tau` is the map that sends the root basis
                 elements `e_{\alpha} \leftrightarrow e_{-\alpha}` and fixes
                 the Cartan subalgebra `h_{\alpha}`. It is an anti-involution
-                in the sense `[\tau(a), \tau(b)] = \tau([a, b])`.
+                in the sense `[\tau(a), \tau(b)] = \tau([b, a])`.
 
                 EXAMPLES::
 

src/sage/combinat/root_system/root_lattice_realizations.py

@@ -3922,8 +3922,8 @@ def is_verma_dominant(self, positive=True):
 
                 \langle \lambda + \rho, \alpha^{\vee} \rangle \notin \ZZ_{<0}
 
-            for all positive roots `\alpha`. Note that this does *not*
-            imply that `\langle \lambda + \rho, \alpha^{\vee} \rangle \geq 0`
+            for all positive roots `\alpha`. Note that begin Verma dominant does
+            *not* imply that `\langle \lambda+\rho, \alpha^{\vee} \rangle \geq 0`
             for any positive root `\alpha`. This is used to determine if
             a Verma module is simple or projective.
 

src/sage/combinat/root_system/weight_space.py

@@ -530,7 +530,7 @@ def is_dominant(self):
             sage: w.is_dominant()
             False
 
-        In the extended affine weight lattice, 'delta' is orthogonal to
+        In the extended affine weight lattice, ``'delta'`` is orthogonal to
         the positive coroots, so adding or subtracting it should not
         affect dominance::
 
@@ -563,7 +563,7 @@ def is_dominant_weight(self):
             sage: w.is_dominant_weight()
             False
 
-        In the extended affine weight lattice, 'delta' is orthogonal to
+        In the extended affine weight lattice, ``'delta'`` is orthogonal to
         the positive coroots, so adding or subtracting it should not
         affect dominance::
 

src/sage/monoids/indexed_free_monoid.py

@@ -291,7 +291,7 @@ def support(self):
         try:
             return sorted(supp, key=print_options['sorting_key'],
                           reverse=print_options['sorting_reverse'])
-        except Exception: # Sorting the output is a plus, but if we can't, no big deal
+        except Exception:  # Sorting the output is a plus, but if we can't, no big deal
             return list(supp)
 
     def leading_support(self):
@@ -527,7 +527,7 @@ def _sorted_items(self):
         try:
             v.sort(key=print_options['sorting_key'],
                    reverse=print_options['sorting_reverse'])
-        except Exception: # Sorting the output is a plus, but if we can't, no big deal
+        except Exception:  # Sorting the output is a plus, but if we can't, no big deal
             pass
         return v
 
@@ -967,7 +967,7 @@ class IndexedFreeAbelianMonoid(IndexedMonoid):
     .. TODO::
 
         Implement a subclass when the index sets is finite that utilizes
-        vectors or the polydict monomials with the index order is fixed.
+        vectors or the polydict monomials with the index order fixed.
     """
     def _repr_(self):
         """