diff --git a/src/sage/combinat/root_system/ambient_space.py b/src/sage/combinat/root_system/ambient_space.py
index 5fe5e161575..0d00c45273f 100644
--- a/src/sage/combinat/root_system/ambient_space.py
+++ b/src/sage/combinat/root_system/ambient_space.py
@@ -369,8 +369,7 @@ def _repr_(self):
 
     def inner_product(self, lambdacheck):
         """
-        The scalar product with elements of the coroot lattice
-        embedded in the ambient space.
+        The scalar product with elements of the ambient space.
 
         EXAMPLES::
 
@@ -379,15 +378,32 @@ def inner_product(self, lambdacheck):
             (-1, 0, 0)
             sage: a.inner_product(a)
             2
+
+        TESTS:
+
+        Verify that :issue:`15325` (A) is fixed::
+
+            sage: rt = RootSystem(['E', 8])
+            sage: lat = rt.root_lattice()
+            sage: spc = rt.ambient_space()
+            sage: spc.simple_root(1).scalar(lat.simple_coroot(2))
+            0
         """
+        if self.parent() is not lambdacheck.parent():
+            try:
+                # see if lambdacheck can be converted to the ambient space
+                lambdacheck = self.parent()(lambdacheck)
+            except (TypeError, ValueError):
+                raise TypeError(f"unable to coerce {lambdacheck} into {self.parent()}")
+
+        # self and lambdacheck both belong to the same ambient space, so use the inner product there
         self_mc = self._monomial_coefficients
         lambdacheck_mc = lambdacheck._monomial_coefficients
 
         result = self.parent().base_ring().zero()
         for t,c in lambdacheck_mc.items():
-            if t not in self_mc:
-                continue
-            result += c*self_mc[t]
+            if t in self_mc:
+                result += c*self_mc[t]
         return result
 
     scalar = inner_product
diff --git a/src/sage/combinat/root_system/root_space.py b/src/sage/combinat/root_system/root_space.py
index 96a68eacc0f..14f5ed4ab23 100644
--- a/src/sage/combinat/root_system/root_space.py
+++ b/src/sage/combinat/root_system/root_space.py
@@ -260,13 +260,34 @@ def scalar(self, lambdacheck):
             [-1  2 -1  0]
             [ 0 -1  2 -1]
             [ 0  0 -2  2]
+
+        TESTS:
+
+        Verify that :issue:`15325` (A) is fixed::
+
+            sage: rt = RootSystem(['E', 8])
+            sage: lat = rt.root_lattice()
+            sage: spc = rt.ambient_space()
+            sage: lat.simple_root(1).scalar(spc.simple_coroot(2))
+            0
+
+        Verify that directionality is correct for roots of different lengths::
+
+            sage: lat = RootSystem(['B', 3]).root_lattice()
+            sage: lat.simple_root(2).scalar(lat.simple_coroot(3))
+            -2
         """
-        # Find some better test
-        if not (lambdacheck in self.parent().coroot_lattice() or lambdacheck in self.parent().coroot_space()):
-            raise TypeError("%s is not in a coroot lattice/space" % (lambdacheck))
-        zero = self.parent().base_ring().zero()
-        cartan_matrix = self.parent().dynkin_diagram()
-        return sum( (sum( (lambdacheck[i]*s for i,s in cartan_matrix.column(j)), zero) * c for j,c in self), zero)
+        if lambdacheck in self.parent().coroot_lattice() or lambdacheck in self.parent().coroot_space():
+            # This is the mathematically canonical case, where we use the Cartan matrix to find the scalar product
+            zero = self.parent().base_ring().zero()
+            cartan_matrix = self.parent().dynkin_diagram()
+            return sum( (sum( (lambdacheck[i]*s for i,s in cartan_matrix.column(j)), zero) * c for j,c in self), zero)
+
+        if lambdacheck in self.parent().root_system.ambient_space():
+            # lambdacheck lives in the ambient space of the root space, so we take the usual dot product in the ambient space
+            return self.to_ambient().dot_product(lambdacheck)
+
+        raise TypeError(f"{lambdacheck} is not in a coroot lattice/space")
 
     def is_positive_root(self):
         """