diff --git a/src/group_operations/multiplication_operation.jl b/src/group_operations/multiplication_operation.jl
index 26e93e4..c309144 100644
--- a/src/group_operations/multiplication_operation.jl
+++ b/src/group_operations/multiplication_operation.jl
@@ -14,7 +14,6 @@ A group operation that is realised by a matrix multiplication.
 """
 struct MatrixMultiplicationGroupOperation <: AbstractMultiplicationGroupOperation end
 
-Base.:*(e::Identity{<:AbstractMultiplicationGroupOperation}) = e
 Base.:*(::Identity{MatrixMultiplicationGroupOperation}, p::Union{AbstractMatrix,Number}) = p
 function Base.:*(
     p::Union{AbstractMatrix,Number}, ::Identity{MatrixMultiplicationGroupOperation}
@@ -28,12 +27,12 @@ function Base.:*(
     return e
 end
 function Base.:*(
-    ::Identity{<:AbstractMultiplicationGroupOperation}, e::Identity{AdditionGroupOperation}
+    e::Identity{<:AbstractMultiplicationGroupOperation}, ::Identity{AdditionGroupOperation}
 )
     return e
 end
 function Base.:*(
-    e::Identity{AdditionGroupOperation}, ::Identity{<:AbstractMultiplicationGroupOperation}
+    ::Identity{AdditionGroupOperation}, e::Identity{<:AbstractMultiplicationGroupOperation}
 )
     return e
 end
diff --git a/test/operations/test_multiplication_operation.jl b/test/operations/test_multiplication_operation.jl
index 45a51d3..634782c 100644
--- a/test/operations/test_multiplication_operation.jl
+++ b/test/operations/test_multiplication_operation.jl
@@ -16,9 +16,11 @@ using LieGroups, Test
         @test (e \ e) === e
         @test inv(e) === e
         @test det(e)
+        ea = Identity(AdditionGroupOperation)
+        @test ea * e === e
+        @test e * ea === e
         # Zero array
         g2 = fill(2.0, ())
-
         # Array
         g3 = [2.0 0.0; 0.0 2.0]
     end