diff --git a/src/sage/algebras/exterior_algebra_groebner.pyx b/src/sage/algebras/exterior_algebra_groebner.pyx
index 265e9323536..3afc924e7ae 100644
--- a/src/sage/algebras/exterior_algebra_groebner.pyx
+++ b/src/sage/algebras/exterior_algebra_groebner.pyx
@@ -477,13 +477,30 @@ cdef class GroebnerStrategy:
             0
             sage: I._groebner_strategy.reduce(E.zero())
             0
+
+        Check :issue:`37108` is fixed::
+
+            sage: E = ExteriorAlgebra(QQ, 6)
+            sage: E.inject_variables(verbose=False)
+            sage: gens = [-e0*e1*e2 + e0*e1*e5 - e0*e2*e3 - e0*e3*e5 + e1*e2*e3 + e1*e3*e5,
+            ....:         e1*e2 - e1*e5 + e2*e5, e0*e2 - e0*e4 + e2*e4,
+            ....:         e3*e4 - e3*e5 + e4*e5, e0*e1 - e0*e3 + e1*e3]
+            sage: I = E.ideal(gens)
+            sage: S = E.quo(I)
+            sage: I.reduce(e1*e3*e4*e5)
+            0
         """
         if not f:
             return f
         # Make a copy to mutate
         f = type(f)(f._parent, copy(f._monomial_coefficients))
-        for g in self.groebner_basis:
-            self.reduce_single(f, g)
+        was_reduced = True
+        while was_reduced:
+            was_reduced = False
+            for g in self.groebner_basis:
+                was_reduced = self.reduce_single(f, g)
+                if was_reduced:
+                    break
         return f
 
     cdef bint reduce_single(self, CliffordAlgebraElement f, CliffordAlgebraElement g) except -1: