diff --git a/src/Bridges/Variable/kernel.jl b/src/Bridges/Variable/kernel.jl
index fb83a7ad7..def2a0022 100644
--- a/src/Bridges/Variable/kernel.jl
+++ b/src/Bridges/Variable/kernel.jl
@@ -19,8 +19,12 @@ function MOI.Bridges.Variable.bridge_constrained_variable(
         gram, vars, con = SOS.add_gram_matrix(model, M, gram_basis, T)
         push!(variables, vars)
         push!(constraints, con)
-        MA.operate_to!(cache, +, gram)
-        MA.operate!(SA.UnsafeAddMul(*), acc, cache, weight)
+        if isone(weight)
+            MA.operate!(SA.UnsafeAdd(), acc, gram)
+        else
+            MA.operate_to!(cache, +, gram)
+            MA.operate!(SA.UnsafeAddMul(*), acc, cache, weight)
+        end
     end
     MA.operate!(SA.canonical, SA.coeffs(acc))
     return KernelBridge{T,M}(
diff --git a/src/Certificate/ideal.jl b/src/Certificate/ideal.jl
index 4def1276a..3e1f7fe0e 100644
--- a/src/Certificate/ideal.jl
+++ b/src/Certificate/ideal.jl
@@ -37,12 +37,22 @@ function _combine_with_gram(
         )
     end
     for (gram, weight) in zip(gram_bases, weights)
-        MA.operate_to!(
-            cache,
-            +,
-            GramMatrix{_NonZero}((_, _) -> _NonZero(), gram),
-        )
-        MA.operate!(SA.UnsafeAddMul(*), p, cache, weight)
+        if isone(weight)
+            MA.operate!(
+                SA.UnsafeAdd(),
+                p,
+                SA.QuadraticForm(
+                    GramMatrix{_NonZero}((_, _) -> _NonZero(), gram),
+                ),
+            )
+        else
+            MA.operate_to!(
+                cache,
+                +,
+                GramMatrix{_NonZero}((_, _) -> _NonZero(), gram),
+            )
+            MA.operate!(SA.UnsafeAddMul(*), p, cache, weight)
+        end
     end
     MA.operate!(SA.canonical, SA.coeffs(p))
     return SA.sub_basis(parent(basis), keys(SA.coeffs(p)))
diff --git a/test/Bridges/Variable/kernel.jl b/test/Bridges/Variable/kernel.jl
index e82aa67b9..45acc0ab4 100644
--- a/test/Bridges/Variable/kernel.jl
+++ b/test/Bridges/Variable/kernel.jl
@@ -68,6 +68,105 @@ function test_runtests()
     return
 end
 
+function test_runtests_weighted()
+    @polyvar x y
+    MOI.Bridges.runtests(
+        SumOfSquares.Bridges.Variable.KernelBridge,
+        model -> begin
+            p, _ = MOI.add_constrained_variables(
+                model,
+                SumOfSquares.WeightedSOSCone{
+                    MOI.PositiveSemidefiniteConeTriangle,
+                }(
+                    MB.SubBasis{MB.Monomial}([
+                        y^4,
+                        x * y^3,
+                        x^2 * y^2,
+                        x^3 * y,
+                        x^4,
+                    ]),
+                    [MB.SubBasis{MB.Monomial}([y^2, x * y, x^2])],
+                    [MB.algebra_element(2.0 * x^0 * y^0)],
+                ),
+            )
+            a = float.(1:length(p))
+            MOI.add_constraint(
+                model,
+                MOI.Utilities.vectorize([a' * p]),
+                MOI.Zeros(1),
+            )
+        end,
+        model -> begin
+            q, _ = MOI.add_constrained_variables(
+                model,
+                MOI.PositiveSemidefiniteConeTriangle(3),
+            )
+            a = float.(1:length(q))
+            MOI.add_constraint(
+                model,
+                MOI.Utilities.vectorize([
+                    2.0 * q[1] +
+                    8.0 * q[2] +
+                    6.0 * (1.0q[3] + 2.0q[4]) +
+                    16.0 * q[5] +
+                    10.0 * q[6],
+                ]),
+                MOI.Zeros(1),
+            )
+        end;
+        cannot_unbridge = true,
+    )
+    return
+end
+
+# weight (1 + x), gram basis [1, x]
+# (1 + x) * (Q11 + 2*Q12*x + Q22*x^2)
+#   = Q11 + (2*Q12 + Q11)*x + (Q22 + 2*Q12)*x^2 + Q22*x^3
+function test_runtests_polynomial_weight()
+    @polyvar x
+    MOI.Bridges.runtests(
+        SumOfSquares.Bridges.Variable.KernelBridge,
+        model -> begin
+            p, _ = MOI.add_constrained_variables(
+                model,
+                SumOfSquares.WeightedSOSCone{
+                    MOI.PositiveSemidefiniteConeTriangle,
+                }(
+                    MB.SubBasis{MB.Monomial}([x^0, x, x^2, x^3]),
+                    [MB.SubBasis{MB.Monomial}([x^0, x])],
+                    [MB.algebra_element(1.0 * x^0 + 1.0 * x)],
+                ),
+            )
+            a = float.(1:length(p))
+            MOI.add_constraint(
+                model,
+                MOI.Utilities.vectorize([a' * p]),
+                MOI.Zeros(1),
+            )
+        end,
+        model -> begin
+            q, _ = MOI.add_constrained_variables(
+                model,
+                SumOfSquares.PositiveSemidefinite2x2ConeTriangle(),
+            )
+            # p[1] = 1.0*q[1], p[2] = 1.0*q[1] + 2.0*q[2],
+            # p[3] = 2.0*q[2] + 1.0*q[3], p[4] = 1.0*q[3]
+            MOI.add_constraint(
+                model,
+                MOI.Utilities.vectorize([
+                    1.0 * q[1] +
+                    2.0 * (1.0q[1] + 2.0q[2]) +
+                    3.0 * (2.0q[2] + 1.0q[3]) +
+                    4.0 * q[3],
+                ]),
+                MOI.Zeros(1),
+            )
+        end;
+        cannot_unbridge = true,
+    )
+    return
+end
+
 end  # module
 
 TestVariableKernel.runtests()