diff --git a/geometry/BUILD.bazel b/geometry/BUILD.bazel
index 1a5ff47..7e16d3b 100644
--- a/geometry/BUILD.bazel
+++ b/geometry/BUILD.bazel
@@ -7,8 +7,9 @@ cc_library(
         "src/detail/all_same.hpp",
         "src/detail/contract_dimensions.hpp",
         "src/detail/ordered.hpp",
-        "src/detail/sort_dimensions.hpp",
         "src/detail/strictly_increasing.hpp",
+        "src/detail/type_list.hpp",
+        "src/detail/type_sort.hpp",
     ],
     hdrs = ["geometry.hpp"],
     visibility = ["//:__subpackages__"],
diff --git a/geometry/src/algebra.hpp b/geometry/src/algebra.hpp
index 4ad1bb5..95047c9 100644
--- a/geometry/src/algebra.hpp
+++ b/geometry/src/algebra.hpp
@@ -3,8 +3,9 @@
 #include "geometry/src/detail/all_same.hpp"
 #include "geometry/src/detail/contract_dimensions.hpp"
 #include "geometry/src/detail/ordered.hpp"
-#include "geometry/src/detail/sort_dimensions.hpp"
 #include "geometry/src/detail/strictly_increasing.hpp"
+#include "geometry/src/detail/type_list.hpp"
+#include "geometry/src/detail/type_sort.hpp"
 
 #include <cstddef>
 #include <type_traits>
@@ -35,6 +36,16 @@ struct algebra
   template <std::size_t... Is>
   struct blade;
 
+  /// multivector
+  /// @tparam Bs blade types
+  ///
+  /// A linear combination of blades.
+  ///
+  /// @note a multivector may be composed of only blades with the same grade
+  ///
+  template <class... Bs>
+  struct multivector;
+
   /// determines if a type is a blade
   /// @tparam T type
   ///
@@ -50,25 +61,29 @@ struct algebra
   /// @}
 
 private:
-  template <std::size_t... Is>
-  static auto rebind(std::index_sequence<Is...>) -> blade<Is...>;
+  template <class... Is>
+  static auto rebind(detail::type_list<Is...>) -> blade<Is::value...>;
 
   template <class Seq>
   using rebind_t = decltype(rebind(Seq{}));
 
+  template <std::size_t... Is>
+  using sequence_t =
+      detail::type_list<std::integral_constant<std::size_t, Is>...>;
+
   template <std::size_t... Is>
   using reified_blade_t = rebind_t<detail::contract_dimensions_t<
       detail::contraction_map::projective,
-      detail::sort_dimensions_t<std::index_sequence<Is...>>>>;
+      detail::type_sort_t<sequence_t<Is...>>>>;
 
   template <std::size_t... Is>
   static constexpr auto reified_blade_coefficient_v =
-      (detail::sort_dimensions_swap_count_v<std::index_sequence<Is...>> % 2 == 0
+      (detail::type_sort_swap_count_v<sequence_t<Is...>> % 2 == 0
            ? scalar_type{1}
            : -scalar_type{1}) *
       scalar_type{detail::contract_dimensions_coefficient_v<
           detail::contraction_map::projective,
-          detail::sort_dimensions_t<std::index_sequence<Is...>>>};
+          detail::type_sort_t<sequence_t<Is...>>>};
 
 public:
   /// unit blade
@@ -190,21 +205,38 @@ struct algebra
 
     /// addition
     ///
+    /// @{
     [[nodiscard]]
     friend constexpr auto
     operator+(blade x, blade y) -> blade
     {
       return blade{x.coefficient + y.coefficient};
     }
+    template <std::size_t... Js>
+    [[nodiscard]]
+    friend constexpr auto
+    operator+(blade x, blade<Js...> y)
+    {
+      if constexpr (
+          detail::ordered<blade<Js...>>{} < detail::ordered<blade>{}) {
+        return multivector<blade<Js...>, blade>{y, x};
+      } else {
+        return multivector<blade, blade<Js...>>{x, y};
+      }
+    }
+    /// @}
 
     /// subtraction
     ///
+    /// @{
+    template <std::size_t... Js>
     [[nodiscard]]
     friend constexpr auto
-    operator-(blade x, blade y) -> blade
+    operator-(blade x, blade<Js...> y)
     {
-      return blade{x.coefficient - y.coefficient};
+      return x + -y;
     }
+    /// @}
 
     /// scalar multiplication
     ///
@@ -234,13 +266,6 @@ struct algebra
     }
   };
 
-  /// multivector
-  /// @tparam Bs blade types
-  ///
-  /// A linear combination of blades.
-  ///
-  /// @note a multivector may be composed of only blades with the same grade
-  ///
   template <class... Bs>
   struct multivector : Bs...
   {
diff --git a/geometry/src/detail/contract_dimensions.hpp b/geometry/src/detail/contract_dimensions.hpp
index 8229e6d..5a8bb5d 100644
--- a/geometry/src/detail/contract_dimensions.hpp
+++ b/geometry/src/detail/contract_dimensions.hpp
@@ -1,5 +1,7 @@
 #pragma once
 
+#include "geometry/src/detail/type_list.hpp"
+
 #include <cstddef>
 #include <utility>
 
@@ -14,49 +16,27 @@ struct contraction_map
   };
 };
 
-template <
-    class Map,
-    class InSeq,
-    int Coeff = 1,
-    class OutSeq = std::index_sequence<>>
+template <class Map, class InSeq, int Coeff = 1, class OutSeq = type_list<>>
 struct contract_dimensions
 {
   using type = OutSeq;
   static constexpr auto coefficient = Coeff;
 };
-template <
-    class Map,
-    std::size_t I0,
-    std::size_t... Is,
-    int Coeff,
-    std::size_t... Js>
-struct contract_dimensions<
-    Map,
-    std::index_sequence<I0, Is...>,
-    Coeff,
-    std::index_sequence<Js...>>
-    : contract_dimensions<
-          Map,
-          std::index_sequence<Is...>,
-          Coeff,
-          std::index_sequence<Js..., I0>>
+template <class Map, class I0, class... Is, int Coeff, class... Js>
+struct contract_dimensions<Map, type_list<I0, Is...>, Coeff, type_list<Js...>>
+    : contract_dimensions<Map, type_list<Is...>, Coeff, type_list<Js..., I0>>
 {};
-template <
-    class Map,
-    std::size_t I0,
-    std::size_t... Is,
-    int Coeff,
-    std::size_t... Js>
+template <class Map, class I0, class... Is, int Coeff, class... Js>
 struct contract_dimensions<
     Map,
-    std::index_sequence<I0, I0, Is...>,
+    type_list<I0, I0, Is...>,
     Coeff,
-    std::index_sequence<Js...>>
+    type_list<Js...>>
     : contract_dimensions<
           Map,
-          std::index_sequence<Is...>,
-          Coeff * Map::template value<I0>,
-          std::index_sequence<Js...>>
+          type_list<Is...>,
+          Coeff * Map::template value<I0{}>,
+          type_list<Js...>>
 {};
 
 template <class Map, class Seq>
diff --git a/geometry/src/detail/sort_dimensions.hpp b/geometry/src/detail/sort_dimensions.hpp
deleted file mode 100644
index 7fa1ebb..0000000
--- a/geometry/src/detail/sort_dimensions.hpp
+++ /dev/null
@@ -1,68 +0,0 @@
-#pragma once
-
-#include <algorithm>
-#include <cstddef>
-#include <type_traits>
-#include <utility>
-
-namespace geometry::detail {
-
-template <class InSeq, class OutSeq = std::index_sequence<>>
-struct sort_swap_first
-{
-  using type = OutSeq;
-};
-template <std::size_t I0, std::size_t... Js>
-struct sort_swap_first<std::index_sequence<I0>, std::index_sequence<Js...>>
-{
-  using type = std::index_sequence<Js..., I0>;
-};
-template <std::size_t I0, std::size_t I1, std::size_t... Is, std::size_t... Js>
-struct sort_swap_first<
-    std::index_sequence<I0, I1, Is...>,
-    std::index_sequence<Js...>>
-    : std::conditional_t<
-          (I0 > I1),
-          sort_swap_first<std::index_sequence<>,
-                          std::index_sequence<Js..., I1, I0, Is...>>,
-          sort_swap_first<std::index_sequence<I1, Is...>,
-                          std::index_sequence<Js..., I0>>>
-{};
-
-template <class Seq>
-using sort_swap_first_t = typename sort_swap_first<Seq>::type;
-
-template <class Seq, std::size_t Swaps = 0, class = void>
-struct sort_swap : sort_swap<sort_swap_first_t<Seq>, Swaps + 1>
-{};
-template <std::size_t... Is, std::size_t Swaps>
-struct sort_swap<
-    std::index_sequence<Is...>,
-    Swaps,
-    std::enable_if_t<std::is_same_v<
-        std::index_sequence<Is...>,
-        sort_swap_first_t<std::index_sequence<Is...>>>>>
-{
-  using type = std::index_sequence<Is...>;
-  static constexpr auto swap_count = Swaps;
-};
-
-template <class Seq>
-struct sort_dimensions
-{
-  using type = typename sort_swap<Seq>::type;
-};
-
-template <class Seq>
-using sort_dimensions_t = typename sort_dimensions<Seq>::type;
-
-template <class Seq>
-struct sort_dimensions_swap_count
-    : std::integral_constant<std::size_t, sort_swap<Seq>::swap_count>
-{};
-
-template <class Seq>
-inline constexpr auto sort_dimensions_swap_count_v =
-    sort_dimensions_swap_count<Seq>::value;
-
-}  // namespace geometry::detail
diff --git a/geometry/src/detail/type_list.hpp b/geometry/src/detail/type_list.hpp
new file mode 100644
index 0000000..58f7733
--- /dev/null
+++ b/geometry/src/detail/type_list.hpp
@@ -0,0 +1,9 @@
+#pragma once
+
+namespace geometry::detail {
+
+template <class...>
+struct type_list
+{};
+
+}  // namespace geometry::detail
diff --git a/geometry/src/detail/type_sort.hpp b/geometry/src/detail/type_sort.hpp
new file mode 100644
index 0000000..274c37b
--- /dev/null
+++ b/geometry/src/detail/type_sort.hpp
@@ -0,0 +1,64 @@
+#pragma once
+
+#include "geometry/src/detail/type_list.hpp"
+
+#include <cstddef>
+#include <type_traits>
+#include <utility>
+
+namespace geometry::detail {
+
+template <class InSeq, class OutSeq = type_list<>>
+struct sort_swap_first
+{
+  using type = OutSeq;
+};
+
+template <class I0, class... Js>
+struct sort_swap_first<type_list<I0>, type_list<Js...>>
+{
+  using type = type_list<Js..., I0>;
+};
+template <class I0, class I1, class... Is, class... Js>
+struct sort_swap_first<type_list<I0, I1, Is...>, type_list<Js...>>
+    : std::conditional_t<
+          (I0{} > I1{}),
+          sort_swap_first<type_list<>, type_list<Js..., I1, I0, Is...>>,
+          sort_swap_first<type_list<I1, Is...>, type_list<Js..., I0>>>
+{};
+
+template <class Seq>
+using sort_swap_first_t = typename sort_swap_first<Seq>::type;
+
+template <class Seq, std::size_t Swaps = 0, class = void>
+struct sort_swap : sort_swap<sort_swap_first_t<Seq>, Swaps + 1>
+{};
+template <class... Is, std::size_t Swaps>
+struct sort_swap<
+    type_list<Is...>,
+    Swaps,
+    std::enable_if_t<
+        std::is_same_v<type_list<Is...>, sort_swap_first_t<type_list<Is...>>>>>
+{
+  using type = type_list<Is...>;
+  static constexpr auto swap_count = Swaps;
+};
+
+template <class Seq>
+struct type_sort
+{
+  using type = typename sort_swap<Seq>::type;
+};
+
+template <class Seq>
+using type_sort_t = typename type_sort<Seq>::type;
+
+template <class Seq>
+struct type_sort_swap_count
+    : std::integral_constant<std::size_t, sort_swap<Seq>::swap_count>
+{};
+
+template <class Seq>
+inline constexpr auto type_sort_swap_count_v = type_sort_swap_count<Seq>::value;
+
+}  // namespace geometry::detail
diff --git a/test/algebra_multivector_test.cpp b/test/algebra_multivector_test.cpp
index cabeaa8..af478bd 100644
--- a/test/algebra_multivector_test.cpp
+++ b/test/algebra_multivector_test.cpp
@@ -45,4 +45,12 @@ auto main() -> int
   "multivector with same elements comparable"_ctest *
       param_ref<multivectors> =  //
       [](auto v) { return expect(eq(v, v)); };
+
+  "multivector constructible from blade addition"_ctest =  //
+      [] {
+        return expect(
+            eq(multivector(2 * e<0>, 3 * e<1>), 2 * e<0> + 3 * e<1>) and
+            eq(multivector(2 * e<0>, 3 * e<1>), 3 * e<1> + 2 * e<0>) and
+            eq(multivector(-2 * e<0>, 3 * e<1>), 3 * e<1> - 2 * e<0>));
+      };
 }