diff --git a/src/arithmetic.jl b/src/arithmetic.jl
index ad8c1f3..3ff2693 100644
--- a/src/arithmetic.jl
+++ b/src/arithmetic.jl
@@ -24,20 +24,28 @@ end
 for op in [:+, :-]
     @eval begin
         function MA.promote_operation(::typeof($op), ::Type{P}, ::Type{Q}) where {P,Q<:_AE}
+            I = MA.promote_operation(implicit, Q)
             return MA.promote_operation(
                 $op,
-                constant_algebra_element_type(MA.promote_operation(SA.basis, Q), P),
-                Q,
+                constant_algebra_element_type(MA.promote_operation(SA.basis, I), P),
+                I,
             )
         end
-        Base.$op(p, q::_AE) = $op(constant_algebra_element(typeof(SA.basis(q)), p), q)
+        function Base.$op(p, q::_AE)
+            i = implicit(q)
+            return $op(constant_algebra_element(typeof(SA.basis(i)), p), i)
+        end
         function MA.promote_operation(::typeof($op), ::Type{P}, ::Type{Q}) where {P<:_AE,Q}
+            I = MA.promote_operation(implicit, P)
             return MA.promote_operation(
                 $op,
-                P,
-                constant_algebra_element_type(MA.promote_operation(SA.basis, P), Q),
+                I,
+                constant_algebra_element_type(MA.promote_operation(SA.basis, I), Q),
             )
         end
-        Base.$op(p::_AE, q) = $op(p, constant_algebra_element(typeof(SA.basis(p)), q))
+        function Base.$op(p::_AE, q)
+            i = implicit(p)
+            return $op(i, constant_algebra_element(typeof(SA.basis(i)), q))
+        end
     end
 end
diff --git a/src/monomial.jl b/src/monomial.jl
index 714bcd1..c6de2d9 100644
--- a/src/monomial.jl
+++ b/src/monomial.jl
@@ -127,6 +127,21 @@ end
 implicit_basis(::SubBasis{B,M}) where {B,M} = FullBasis{B,M}()
 implicit_basis(basis::FullBasis) = basis
 
+function implicit(a::SA.AlgebraElement)
+    basis = implicit_basis(SA.basis(a))
+    return algebra_element(SA.coeffs(a, basis), basis)
+end
+
+function MA.promote_operation(
+    ::typeof(implicit),
+    ::Type{E},
+) where {AG,T,E<:SA.AlgebraElement{AG,T}}
+    BT = MA.promote_operation(implicit_basis, MA.promote_operation(SA.basis, E))
+    A = MA.promote_operation(algebra, BT)
+    M = MP.monomial_type(BT)
+    return SA.AlgebraElement{A,T,SA.SparseCoefficients{M,T,Vector{M},Vector{T}}}
+end
+
 function MA.promote_operation(
     ::typeof(implicit_basis),
     ::Type{<:Union{FullBasis{B,M},SubBasis{B,M}}},
@@ -393,14 +408,25 @@ function MP.polynomial_type(::Type{FullBasis{B,M}}, ::Type{T}) where {T,B,M}
     return MP.polynomial_type(M, _promote_coef(T, B))
 end
 
+_vec(v::Vector) = v
+_vec(v::AbstractVector) = collect(v)
+
 # Adapted from SA to incorporate `_promote_coef`
 function SA.coeffs(
     cfs,
-    source::MonomialIndexedBasis{B},
-    target::MonomialIndexedBasis{Monomial},
-) where {B}
+    source::MonomialIndexedBasis{B1},
+    target::MonomialIndexedBasis{B2},
+) where {B1,B2}
     source === target && return cfs
     source == target && return cfs
-    res = SA.zero_coeffs(_promote_coef(valtype(cfs), B), target)
-    return SA.coeffs!(res, cfs, source, target)
+    if B1 === B2 && target isa FullBasis
+        # The defaults initialize to zero and then sums which promotes
+        # `JuMP.VariableRef` to `JuMP.AffExpr`
+        return SA.SparseCoefficients(_vec(source.monomials), _vec(cfs))
+    elseif B2 === Monomial
+        res = SA.zero_coeffs(_promote_coef(valtype(cfs), B1), target)
+        return SA.coeffs!(res, cfs, source, target)
+    else
+        error("Convertion from $source to $target not implemented yet")
+    end
 end
diff --git a/test/runtests.jl b/test/runtests.jl
index 23c3455..31cc81a 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -7,9 +7,9 @@ const MB = MultivariateBases
 using LinearAlgebra
 using DynamicPolynomials
 
-function _test_op(op, a, b)
-    result = @inferred op(a, b)
-    @test typeof(result) == MA.promote_operation(op, typeof(a), typeof(b))
+function _test_op(op, args...)
+    result = @inferred op(args...)
+    @test typeof(result) == MA.promote_operation(op, typeof.(args)...)
     return result
 end
 
@@ -53,6 +53,8 @@ function api_test(B::Type{<:MB.AbstractMonomialIndexed}, degree)
         @test length(empty_basis(typeof(basis))) == 0
         @test polynomial_type(basis, Float64) == polynomial_type(x[1], Float64)
         #@test polynomial(i -> 0.0, basis) isa polynomial_type(basis, Float64)
+        a = MB.algebra_element(ones(length(basis)), basis)
+        _test_op(MB.implicit, a)
     end
     mono = x[1]^2 * x[2]^3
     p = MB.Polynomial{B}(mono)