diff --git a/Eigen/src/Core/Product.h b/Eigen/src/Core/Product.h
index 23c90c223..6bad832e0 100644
--- a/Eigen/src/Core/Product.h
+++ b/Eigen/src/Core/Product.h
@@ -21,7 +21,7 @@ class ProductImpl;
 namespace internal {
 
 template <typename Lhs, typename Rhs, int Option>
-struct traits<Product<Lhs, Rhs, Option>> {
+struct traits<Product<Lhs, Rhs, Option> > {
   typedef remove_all_t<Lhs> LhsCleaned;
   typedef remove_all_t<Rhs> RhsCleaned;
   typedef traits<LhsCleaned> LhsTraits;
@@ -93,23 +93,6 @@ class Product
   typedef internal::remove_all_t<LhsNested> LhsNestedCleaned;
   typedef internal::remove_all_t<RhsNested> RhsNestedCleaned;
 
- private:
-  using LhsTransposeType = Transpose<const LhsNestedCleaned>;
-  using LhsScalar = typename internal::traits<LhsNestedCleaned>::Scalar;
-  using LhsConjugateTransposeType = CwiseUnaryOp<internal::scalar_conjugate_op<LhsScalar>, LhsTransposeType>;
-  using LhsAdjointType =
-      std::conditional_t<is_complex_helper<LhsScalar>::value, LhsConjugateTransposeType, LhsTransposeType>;
-
-  using RhsTransposeType = Transpose<const RhsNestedCleaned>;
-  using RhsScalar = typename internal::traits<RhsNestedCleaned>::Scalar;
-  using RhsConjugateTransposeType = CwiseUnaryOp<internal::scalar_conjugate_op<RhsScalar>, RhsTransposeType>;
-  using RhsAdjointType =
-      std::conditional_t<is_complex_helper<RhsScalar>::value, RhsConjugateTransposeType, RhsTransposeType>;
-
- public:
-  using TransposeReturnType = Product<RhsTransposeType, LhsTransposeType, Option>;
-  using AdjointReturnType = Product<RhsAdjointType, LhsAdjointType, Option>;
-
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Product(const Lhs& lhs, const Rhs& rhs) : m_lhs(lhs), m_rhs(rhs) {
     eigen_assert(lhs.cols() == rhs.rows() && "invalid matrix product" &&
                  "if you wanted a coeff-wise or a dot product use the respective explicit functions");
@@ -121,9 +104,6 @@ class Product
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const LhsNestedCleaned& lhs() const { return m_lhs; }
   EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE const RhsNestedCleaned& rhs() const { return m_rhs; }
 
-  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE TransposeReturnType transpose() const;
-  EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE AdjointReturnType adjoint() const;
-
  protected:
   LhsNested m_lhs;
   RhsNested m_rhs;
@@ -132,12 +112,12 @@ class Product
 namespace internal {
 
 template <typename Lhs, typename Rhs, int Option, int ProductTag = internal::product_type<Lhs, Rhs>::ret>
-class dense_product_base : public internal::dense_xpr_base<Product<Lhs, Rhs, Option>>::type {};
+class dense_product_base : public internal::dense_xpr_base<Product<Lhs, Rhs, Option> >::type {};
 
 /** Conversion to scalar for inner-products */
 template <typename Lhs, typename Rhs, int Option>
 class dense_product_base<Lhs, Rhs, Option, InnerProduct>
-    : public internal::dense_xpr_base<Product<Lhs, Rhs, Option>>::type {
+    : public internal::dense_xpr_base<Product<Lhs, Rhs, Option> >::type {
   typedef Product<Lhs, Rhs, Option> ProductXpr;
   typedef typename internal::dense_xpr_base<ProductXpr>::type Base;
 
diff --git a/Eigen/src/Core/Transpose.h b/Eigen/src/Core/Transpose.h
index f12cf4a8d..1cc7a2867 100644
--- a/Eigen/src/Core/Transpose.h
+++ b/Eigen/src/Core/Transpose.h
@@ -196,18 +196,6 @@ EIGEN_DEVICE_FUNC inline const typename MatrixBase<Derived>::AdjointReturnType M
   return AdjointReturnType(this->transpose());
 }
 
-template <typename Lhs_, typename Rhs_, int Option>
-EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE typename Product<Lhs_, Rhs_, Option>::TransposeReturnType
-Product<Lhs_, Rhs_, Option>::transpose() const {
-  return TransposeReturnType(m_rhs.transpose(), m_lhs.transpose());
-}
-
-template <typename Lhs_, typename Rhs_, int Option>
-EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE typename Product<Lhs_, Rhs_, Option>::AdjointReturnType
-Product<Lhs_, Rhs_, Option>::adjoint() const {
-  return AdjointReturnType(m_rhs.adjoint(), m_lhs.adjoint());
-}
-
 /***************************************************************************
  * "in place" transpose implementation
  ***************************************************************************/
diff --git a/Eigen/src/Core/util/XprHelper.h b/Eigen/src/Core/util/XprHelper.h
index ea4a7c3fb..555faa1cc 100644
--- a/Eigen/src/Core/util/XprHelper.h
+++ b/Eigen/src/Core/util/XprHelper.h
@@ -1011,16 +1011,6 @@ struct ScalarBinaryOpTraits<void, void, BinaryOp> {
   typedef void ReturnType;
 };
 
-template <typename Scalar>
-struct is_complex_helper {
-  static constexpr bool value = NumTraits<Scalar>::IsComplex;
-};
-// NumTraits<void> is not defined by intent
-template <>
-struct is_complex_helper<void> {
-  static constexpr bool value = false;
-};
-
 // We require Lhs and Rhs to have "compatible" scalar types.
 // It is tempting to always allow mixing different types but remember that this is often impossible in the vectorized
 // paths. So allowing mixing different types gives very unexpected errors when enabling vectorization, when the user
diff --git a/test/product_notemporary.cpp b/test/product_notemporary.cpp
index f41a4b171..77a296966 100644
--- a/test/product_notemporary.cpp
+++ b/test/product_notemporary.cpp
@@ -66,8 +66,6 @@ void product_notemporary(const MatrixType& m) {
   VERIFY_EVALUATION_COUNT(m3 = (m1 * m2.adjoint()), 1);
   VERIFY_EVALUATION_COUNT(m3 = (m1 * m2.adjoint()).transpose(), 1);
   VERIFY_EVALUATION_COUNT(m3.noalias() = m1 * m2.adjoint(), 0);
-  VERIFY_EVALUATION_COUNT(m3.noalias() = (m1 * m2.adjoint()).transpose(), 0);
-  VERIFY_EVALUATION_COUNT(m3.noalias() = (m1 * m2.transpose()).adjoint(), 0);
 
   VERIFY_EVALUATION_COUNT(m3 = s1 * (m1 * m2.transpose()), 1);
   //   VERIFY_EVALUATION_COUNT( m3 = m3 + s1 * (m1 * m2.transpose()), 1);