diff --git a/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl b/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl
index 28d56ca6..46b4f953 100644
--- a/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl
+++ b/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl
@@ -858,7 +858,7 @@ for (f!, f, adj) in (
         (:project_antihermitian!, :project_antihermitian, :project_antihermitian_adjoint),
     )
     @eval begin
-        @is_primitive DefaultCtx Mooncake.ReverseMode Tuple{typeof($f!), Any, Any, MatrixAlgebraKit.AbstractAlgorithm}
+        @is_primitive DefaultCtx Tuple{typeof($f!), Any, Any, MatrixAlgebraKit.AbstractAlgorithm}
         function Mooncake.rrule!!(f_df::CoDual{typeof($f!)}, A_dA::CoDual, arg_darg::CoDual, alg_dalg::CoDual{<:MatrixAlgebraKit.AbstractAlgorithm})
             A, dA = arrayify(A_dA)
             arg, darg = A_dA === arg_darg ? (A, dA) : arrayify(arg_darg)
@@ -879,8 +879,14 @@ for (f!, f, adj) in (
 
             return arg_darg, $adj
         end
-
-        @is_primitive DefaultCtx Mooncake.ReverseMode Tuple{typeof($f), Any, MatrixAlgebraKit.AbstractAlgorithm}
+        function Mooncake.frule!!(f_df::Dual{typeof($f!)}, A_dA::Dual, arg_darg::Dual, alg_dalg::Dual{<:MatrixAlgebraKit.AbstractAlgorithm})
+            A, dA = arrayify(A_dA)
+            arg, darg = A_dA === arg_darg ? (A, dA) : arrayify(arg_darg)
+            $f!(A, arg, Mooncake.primal(alg_dalg))
+            $f!(dA, darg, Mooncake.primal(alg_dalg))
+            return arg_darg
+        end
+        @is_primitive DefaultCtx Tuple{typeof($f), Any, MatrixAlgebraKit.AbstractAlgorithm}
         function Mooncake.rrule!!(f_df::CoDual{typeof($f)}, A_dA::CoDual, alg_dalg::CoDual{<:MatrixAlgebraKit.AbstractAlgorithm})
             A, dA = arrayify(A_dA)
             output = $f(A, Mooncake.primal(alg_dalg))
@@ -896,6 +902,14 @@ for (f!, f, adj) in (
 
             return output_doutput, $adj
         end
+        function Mooncake.frule!!(f_df::Dual{typeof($f)}, A_dA::Dual, alg_dalg::Dual{<:MatrixAlgebraKit.AbstractAlgorithm})
+            A, dA = arrayify(A_dA)
+            output = $f(A, Mooncake.primal(alg_dalg))
+            output_doutput = Mooncake.zero_dual(output)
+            doutput = last(arrayify(output_doutput))
+            $f!(dA, doutput, Mooncake.primal(alg_dalg))
+            return output_doutput
+        end
     end
 end
 
diff --git a/test/testsuite/enzyme/projections.jl b/test/testsuite/enzyme/projections.jl
index bc3e7014..2c5240a2 100644
--- a/test/testsuite/enzyme/projections.jl
+++ b/test/testsuite/enzyme/projections.jl
@@ -15,39 +15,45 @@ end
 """
     test_enzyme_project_hermitian(T, sz; rng, atol, rtol)
 
-Test the Enzyme reverse-mode AD rule for `project_hermitian` and its in-place variant.
+Test the Enzyme forward- and reverse-mode AD rule for `project_hermitian` and its in-place variant.
 """
 function test_enzyme_project_hermitian(
         T, sz;
         rng = Random.default_rng(), atol::Real = 0, rtol::Real = precision(T),
         fdm = enzyme_fdm(T)
     )
-    return @testset "project_hermitian reverse: RT $RT, TA $TA" for RT in (Duplicated,), TA in (Duplicated,)
+    return @testset "project_hermitian: RT $RT, TA $TA" for RT in (Duplicated,), TA in (Duplicated,)
         A = instantiate_matrix(T, sz)
         B = instantiate_matrix(T, sz)
         alg = MatrixAlgebraKit.select_algorithm(project_hermitian, A)
         test_reverse(project_hermitian, RT, (A, TA), (alg, Const); atol, rtol, fdm)
         test_reverse(project_hermitian!, RT, (A, TA), (B, TA), (alg, Const); atol, rtol, fdm)
         test_reverse(project_hermitian_inplace!, RT, (A, TA), (alg, Const); atol, rtol, fdm)
+        test_forward(project_hermitian, RT, (A, TA), (alg, Const); atol, rtol, fdm)
+        test_forward(project_hermitian!, RT, (A, TA), (B, TA), (alg, Const); atol, rtol, fdm)
+        test_forward(project_hermitian_inplace!, RT, (A, TA), (alg, Const); atol, rtol, fdm)
     end
 end
 
 """
     test_enzyme_project_antihermitian(T, sz; rng, atol, rtol)
 
-Test the Enzyme reverse-mode AD rule for `project_antihermitian` and its in-place variant.
+Test the Enzyme forward- and reverse-mode AD rule for `project_antihermitian` and its in-place variant.
 """
 function test_enzyme_project_antihermitian(
         T, sz;
         rng = Random.default_rng(), atol::Real = 0, rtol::Real = precision(T),
         fdm = enzyme_fdm(T)
     )
-    return @testset "project_antihermitian reverse: RT $RT, TA $TA" for RT in (Duplicated,), TA in (Duplicated,)
+    return @testset "project_antihermitian: RT $RT, TA $TA" for RT in (Duplicated,), TA in (Duplicated,)
         A = instantiate_matrix(T, sz)
         B = instantiate_matrix(T, sz)
         alg = MatrixAlgebraKit.select_algorithm(project_hermitian, A)
         test_reverse(project_antihermitian, RT, (A, TA), (alg, Const); atol, rtol, fdm)
         test_reverse(project_antihermitian!, RT, (A, TA), (B, TA), (alg, Const); atol, rtol, fdm)
         test_reverse(project_antihermitian_inplace!, RT, (A, TA), (alg, Const); atol, rtol, fdm)
+        test_forward(project_antihermitian, RT, (A, TA), (alg, Const); atol, rtol, fdm)
+        test_forward(project_antihermitian!, RT, (A, TA), (B, TA), (alg, Const); atol, rtol, fdm)
+        test_forward(project_antihermitian_inplace!, RT, (A, TA), (alg, Const); atol, rtol, fdm)
     end
 end
diff --git a/test/testsuite/mooncake/projections.jl b/test/testsuite/mooncake/projections.jl
index 74e22a44..359c9cbe 100644
--- a/test/testsuite/mooncake/projections.jl
+++ b/test/testsuite/mooncake/projections.jl
@@ -15,7 +15,7 @@ end
 """
     test_mooncake_project_hermitian(T, sz; rng, atol, rtol)
 
-Test the Mooncake reverse-mode AD rule for `project_hermitian` and its in-place variant.
+Test the Mooncake forward- and reverse-mode AD rule for `project_hermitian` and its in-place variant.
 """
 function test_mooncake_project_hermitian(
         T, sz;
@@ -27,15 +27,15 @@ function test_mooncake_project_hermitian(
         alg = MatrixAlgebraKit.select_algorithm(project_hermitian, A)
         Mooncake.TestUtils.test_rule(
             rng, project_hermitian, A, alg;
-            mode = Mooncake.ReverseMode, atol, rtol
+            atol, rtol
         )
         Mooncake.TestUtils.test_rule(
             rng, project_hermitian!, A, A, alg;
-            mode = Mooncake.ReverseMode, atol, rtol
+            atol, rtol
         )
         Mooncake.TestUtils.test_rule(
             rng, project_hermitian!, A, B, alg;
-            mode = Mooncake.ReverseMode, atol, rtol
+            atol, rtol
         )
     end
 end
@@ -43,7 +43,7 @@ end
 """
     test_mooncake_project_antihermitian(T, sz; rng, atol, rtol)
 
-Test the Mooncake reverse-mode AD rule for `project_antihermitian` and its in-place variant.
+Test the Mooncake forward- and reverse-mode AD rule for `project_antihermitian` and its in-place variant.
 """
 function test_mooncake_project_antihermitian(
         T, sz;
@@ -55,15 +55,15 @@ function test_mooncake_project_antihermitian(
         alg = MatrixAlgebraKit.select_algorithm(project_hermitian, A)
         Mooncake.TestUtils.test_rule(
             rng, project_antihermitian, A, alg;
-            mode = Mooncake.ReverseMode, atol, rtol
+            atol, rtol
         )
         Mooncake.TestUtils.test_rule(
             rng, project_antihermitian!, A, A, alg;
-            mode = Mooncake.ReverseMode, atol, rtol
+            atol, rtol
         )
         Mooncake.TestUtils.test_rule(
             rng, project_antihermitian!, A, B, alg;
-            mode = Mooncake.ReverseMode, atol, rtol
+            atol, rtol
         )
     end
 end