diff --git a/src/promote.jl b/src/promote.jl
index 5473917..a6942e5 100644
--- a/src/promote.jl
+++ b/src/promote.jl
@@ -1,3 +1,5 @@
+const _PolynomialLike = Union{MP.AbstractPolynomialLike,SA.AlgebraElement}
+
 function SA.promote_bases_with_maps(
     ::FullSpace,
     p::Union{MP.AbstractPolynomialLike,AbstractSemialgebraicSet},
@@ -5,6 +7,10 @@ function SA.promote_bases_with_maps(
     return (FullSpace(), nothing), (p, nothing)
 end
 
+function SA.promote_bases_with_maps(::FullSpace, p::SA.AlgebraElement)
+    return (FullSpace(), nothing), (p, nothing)
+end
+
 function SA.promote_bases_with_maps(
     a::AbstractSemialgebraicSet,
     b::Union{MP.AbstractPolynomialLike,AbstractSemialgebraicSet},
@@ -13,6 +19,22 @@ function SA.promote_bases_with_maps(
     return SA.maybe_promote(a, _a...), SA.maybe_promote(b, _b...)
 end
 
+function _promote_ae(b::SA.AlgebraElement, all_vars)
+    alg = SA.parent(b)
+    new_obj = typeof(SA.object(alg))(all_vars)
+    new_basis, _ = SA.promote_with_map(SA.basis(alg), new_obj, identity)
+    (new_alg, alg_map), _ = SA.promote_bases_with_maps(alg, new_basis)
+    return SA.maybe_promote(b, new_alg, alg_map)
+end
+
+function SA.promote_bases_with_maps(
+    a::AbstractSemialgebraicSet,
+    b::SA.AlgebraElement,
+)
+    _a, _b = MP.promote_variables_with_maps(MP.variables(a), MP.variables(b))
+    return SA.maybe_promote(a, _a...), _promote_ae(b, _b[1])
+end
+
 function _map_polys(p, vars, map)
     return first.(SA.promote_with_map.(p, Ref(vars), Ref(map)))
 end
diff --git a/test/Project.toml b/test/Project.toml
index 8903644..9c7700a 100644
--- a/test/Project.toml
+++ b/test/Project.toml
@@ -1,6 +1,7 @@
 [deps]
 DynamicPolynomials = "7c1d4256-1411-5781-91ec-d7bc3513ac07"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+MultivariateBases = "be282fd4-ad43-11e9-1d11-8bd9d7e43378"
 MultivariatePolynomials = "102ac46a-7ee4-5c85-9060-abc95bfdeaa3"
 MutableArithmetics = "d8a4904e-b15c-11e9-3269-09a3773c0cb0"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
diff --git a/test/promote.jl b/test/promote.jl
index d8386d1..cd28c6a 100644
--- a/test/promote.jl
+++ b/test/promote.jl
@@ -2,6 +2,7 @@ using Test
 
 import StarAlgebras as SA
 import MultivariatePolynomials as MP
+import MultivariateBases as MB
 
 using SemialgebraicSets
 
@@ -177,4 +178,55 @@ using SemialgebraicSets
         @test length(eq) == 1
         @test Set(MP.variables(eq[1])) == Set([x, y, z])
     end
+
+    @testset "FullSpace with AlgebraElement" begin
+        f = FullSpace()
+        a = MB.algebra_element(x^2 + y)
+        (new_f, map_f), (new_a, map_a) = SA.promote_bases_with_maps(f, a)
+        @test new_f isa FullSpace
+        @test map_f === nothing
+        @test new_a === a
+        @test map_a === nothing
+    end
+
+    @testset "AlgebraicSet with AlgebraElement - same variables" begin
+        V = @set x * y == 1
+        a = MB.algebra_element(x + y)
+        (new_V, map_V), (new_a, map_a) = SA.promote_bases_with_maps(V, a)
+        @test new_V === V
+        @test map_V === nothing
+        @test new_a === a
+        @test map_a === nothing
+    end
+
+    @testset "AlgebraicSet with AlgebraElement - different variables" begin
+        V = @set x^2 == 1
+        a = MB.algebra_element(y + z)
+        (new_V, map_V), (new_a, map_a) = SA.promote_bases_with_maps(V, a)
+        @test map_V !== nothing
+        @test map_a !== nothing
+        @test Set(MP.variables(new_V)) == Set([x, y, z])
+        @test Set(MP.variables(new_a)) == Set([x, y, z])
+    end
+
+    @testset "BasicSemialgebraicSet with AlgebraElement - same variables" begin
+        S = @set x - y == 0 && x^2 * y >= 1
+        a = MB.algebra_element(x + y)
+        (new_S, map_S), (new_a, map_a) = SA.promote_bases_with_maps(S, a)
+        @test new_S === S
+        @test map_S === nothing
+        @test new_a === a
+        @test map_a === nothing
+    end
+
+    @testset "BasicSemialgebraicSet with AlgebraElement - different variables" begin
+        S = @set x >= 1
+        a = MB.algebra_element(y + z)
+        (new_S, map_S), (new_a, map_a) = SA.promote_bases_with_maps(S, a)
+        @test map_S !== nothing
+        @test map_a !== nothing
+        @test Set(MP.variables(new_a)) == Set([x, y, z])
+        ineqs = inequalities(new_S)
+        @test length(ineqs) == 1
+    end
 end