diff --git a/test/perm_grp_algebra.jl b/test/perm_grp_algebra.jl
index 6d245cb2..b70acd83 100644
--- a/test/perm_grp_algebra.jl
+++ b/test/perm_grp_algebra.jl
@@ -99,4 +99,23 @@
         end
         @test a ≤ b
     end
+
+    @testset "Automatic conversion to explicit" begin
+        a = SA.AlgebraElement([2], SA.StarAlgebra(G, SA.SubBasis([g], db)))
+        b = SA.AlgebraElement([-3], SA.StarAlgebra(G, SA.SubBasis([h], db)))
+        # `Base.+` assumes that using the basis of the first argument will suffice
+        # We should redefine `Base.:+(a::SubBasis, b::SubBasis)` to first
+        # convert `a` and `b` to their implicit basis equivalent and then
+        # do `+` and then convert the result back
+        # `MultivariateBases` defines an `implicit` function.
+        # Why not having an `explicit` as well ?
+        # My dream implementation would be
+        # Base.:+(a::SubBasis, b::SubBasis) = explicit(implicit(a) + implicit(b))
+        # so we just need to implement `implicit` and `explicit` 👼
+        @test_broken SA.explicit(SA.implicit(a)) == a
+        @test_broken SA.explicit(SA.implicit(a)) == a
+        @test_broken SA.explicit(SA.implicit(b)) == b
+        @test_broken a + b == SA.explicit(SA.implicit(a) + SA.implicit(b))
+        @test_broken a * b == SA.explicit(SA.implicit(a) * SA.implicit(b))
+    end
 end