diff --git a/Project.toml b/Project.toml
index 8255f05..75f97da 100644
--- a/Project.toml
+++ b/Project.toml
@@ -19,9 +19,9 @@ Groups = "0.7"
 IntervalArithmetic = "0.20"
 JuMP = "1.3"
 ProgressMeter = "1.7"
-SCS = "1.1.0"
+SCS = "1.1"
 StaticArrays = "1"
-SymbolicWedderburn = "0.3.2"
+SymbolicWedderburn = "0.3.4"
 julia = "1.6"
 
 [extras]
diff --git a/src/PropertyT.jl b/src/PropertyT.jl
index 6b97d5d..16430ee 100644
--- a/src/PropertyT.jl
+++ b/src/PropertyT.jl
@@ -26,13 +26,13 @@ include("gradings.jl")
 
 include("actions/actions.jl")
 
-function group_algebra(G::Groups.Group, S=gens(G); halfradius::Integer, twisted::Bool)
+function group_algebra(G::Groups.Group, S = gens(G); halfradius::Integer)
     S = union!(S, inv.(S))
     @info "generating wl-metric ball of radius $(2halfradius)"
     @time E, sizes = Groups.wlmetric_ball(S, radius=2halfradius)
     @info "sizes = $(sizes)"
     @info "computing the *-algebra structure for G"
-    @time RG = StarAlgebras.StarAlgebra{twisted}(
+    @time RG = StarAlgebras.StarAlgebra(
         G,
         StarAlgebras.Basis{UInt32}(E),
         (sizes[halfradius], sizes[halfradius]),
diff --git a/src/certify.jl b/src/certify.jl
index cdebcc9..2079791 100644
--- a/src/certify.jl
+++ b/src/certify.jl
@@ -8,26 +8,25 @@ end
 function __sos_via_sqr!(
     res::StarAlgebras.AlgebraElement,
     P::AbstractMatrix;
-    augmented::Bool
+    augmented::Bool,
+    id = (b = basis(parent(res)); b[one(first(b))]),
 )
-    StarAlgebras.zero!(res)
     A = parent(res)
-    b = basis(A)
-    @assert size(A.mstructure) == size(P)
-    e = b[one(b[1])]
+    mstr = A.mstructure
+    @assert size(mstr) == size(P)
 
-    for i in axes(A.mstructure, 1)
-        x = StarAlgebras._istwisted(A.mstructure) ? StarAlgebras.star(b[i]) : b[i]
-        for j in axes(A.mstructure, 2)
+    StarAlgebras.zero!(res)
+    for j in axes(mstr, 2)
+        for i in axes(mstr, 1)
             p = P[i, j]
-            xy = b[A.mstructure[i, j]]
-            # either result += P[x,y]*(x*y)
-            res[xy] += p
+            x_star_y = mstr[-i, j]
+            res[x_star_y] += p
+            # either result += P[x,y]*(x'*y)
             if augmented
-                # or result += P[x,y]*(1-x)*(1-y) == P[x,y]*(2-x-y+xy)
-                y = b[j]
-                res[e] += p
-                res[x] -= p
+                # or result += P[x,y]*(1-x)'*(1-y) == P[x,y]*(1-x'-y+x'y)
+                res[id] += p
+                x_star, y = mstr[-i, id], j
+                res[x_star] -= p
                 res[y] -= p
             end
         end
diff --git a/src/constraint_matrix.jl b/src/constraint_matrix.jl
index 5734837..73f3238 100644
--- a/src/constraint_matrix.jl
+++ b/src/constraint_matrix.jl
@@ -132,3 +132,50 @@ function LinearAlgebra.dot(cm::ConstraintMatrix, m::AbstractMatrix{T}) where {T}
     neg = isempty(cm.neg) ? zero(first(m)) : sum(@view m[cm.neg])
     return convert(eltype(cm), cm.val) * (pos - neg)
 end
+
+function constraints(A::StarAlgebras.StarAlgebra; augmented::Bool)
+    return constraints(basis(A), A.mstructure; augmented = augmented)
+end
+
+function constraints(
+    basis::StarAlgebras.AbstractBasis,
+    mstr::StarAlgebras.MultiplicativeStructure;
+    augmented = false,
+)
+    cnstrs = _constraints(
+        mstr;
+        augmented = augmented,
+        num_constraints = length(basis),
+        id = basis[one(first(basis))],
+    )
+
+    return Dict(
+        basis[i] => ConstraintMatrix(c, size(mstr)..., 1) for
+        (i, c) in pairs(cnstrs)
+    )
+end
+
+function _constraints(
+    mstr::StarAlgebras.MultiplicativeStructure;
+    augmented::Bool = false,
+    num_constraints = maximum(mstr),
+    id,
+)
+    cnstrs = [signed(eltype(mstr))[] for _ in 1:num_constraints]
+    LI = LinearIndices(size(mstr))
+
+    for ci in CartesianIndices(size(mstr))
+        k = LI[ci]
+        i, j = Tuple(ci)
+        a_star_b = mstr[-i, j]
+        push!(cnstrs[a_star_b], k)
+        if augmented
+            # (1-a)'(1-b) = 1 - a' - b + a'b
+            push!(cnstrs[id], k)
+            a_star, b = mstr[-i, id], j
+            push!(cnstrs[a_star], -k)
+            push!(cnstrs[b], -k)
+        end
+    end
+    return cnstrs
+end
diff --git a/src/sos_sdps.jl b/src/sos_sdps.jl
index 2f6faea..04a8916 100644
--- a/src/sos_sdps.jl
+++ b/src/sos_sdps.jl
@@ -11,10 +11,8 @@ function sos_problem_dual(
 )
     @assert parent(elt) == parent(order_unit)
     algebra = parent(elt)
-    mstructure = if StarAlgebras._istwisted(algebra.mstructure)
-        algebra.mstructure
-    else
-        StarAlgebras.MTable{true}(basis(algebra), table_size=size(algebra.mstructure))
+    moment_matrix = let m = algebra.mstructure
+        [m[-i, j] for i in axes(m, 1) for j in axes(m, 2)]
     end
 
     # 1 variable for every primal constraint
@@ -24,7 +22,7 @@ function sos_problem_dual(
     model = Model()
     @variable model y[1:length(basis(algebra))]
     @constraint model λ_dual dot(order_unit, y) == 1
-    @constraint(model, psd, y[mstructure] in PSDCone())
+    @constraint(model, psd, y[moment_matrix] in PSDCone())
 
     if !isinf(lower_bound)
         throw("Not Implemented yet")
@@ -37,45 +35,6 @@ function sos_problem_dual(
     return model
 end
 
-function constraints(
-    basis::StarAlgebras.AbstractBasis,
-    mstr::AbstractMatrix{<:Integer};
-    augmented::Bool=false,
-    table_size=size(mstr)
-)
-    cnstrs = [signed(eltype(mstr))[] for _ in basis]
-    LI = LinearIndices(table_size)
-
-    for ci in CartesianIndices(table_size)
-        k = LI[ci]
-        a_star_b = basis[mstr[k]]
-        push!(cnstrs[basis[a_star_b]], k)
-        if augmented
-            # (1-a_star)(1-b) = 1 - a_star - b + a_star_b
-
-            i, j = Tuple(ci)
-            a, b = basis[i], basis[j]
-
-            push!(cnstrs[basis[one(a)]], k)
-            push!(cnstrs[basis[StarAlgebras.star(a)]], -k)
-            push!(cnstrs[basis[b]], -k)
-        end
-    end
-
-    return Dict(
-        basis[i] => ConstraintMatrix(c, table_size..., 1) for (i, c) in pairs(cnstrs)
-    )
-end
-
-function constraints(A::StarAlgebras.StarAlgebra; augmented::Bool, twisted::Bool)
-    mstructure = if StarAlgebras._istwisted(A.mstructure) == twisted
-        A.mstructure
-    else
-        StarAlgebras.MTable{twisted}(basis(A), table_size=size(A.mstructure))
-    end
-    return constraints(basis(A), mstructure, augmented=augmented)
-end
-
 """
     sos_problem_primal(X, [u = zero(X); upper_bound=Inf])
 Formulate sum of squares decomposition problem for `X - λ·u`.
@@ -111,7 +70,7 @@ function sos_problem_primal(
         @warn "Setting `upper_bound` together with zero `order_unit` has no effect"
     end
 
-    A = constraints(parent(elt), augmented=augmented, twisted=true)
+    A = constraints(parent(elt); augmented = augmented)
 
     if !iszero(order_unit)
         λ = JuMP.@variable(model, λ)
@@ -135,9 +94,9 @@ end
 function invariant_constraint!(
     result::AbstractMatrix,
     basis::StarAlgebras.AbstractBasis,
-    cnstrs::AbstractDict{K,CM},
+    cnstrs::AbstractDict{K,<:ConstraintMatrix},
     invariant_vec::SparseVector,
-) where {K,CM<:ConstraintMatrix}
+) where {K}
     result .= zero(eltype(result))
     for i in SparseArrays.nonzeroinds(invariant_vec)
         g = basis[i]
@@ -234,7 +193,7 @@ function sos_problem_primal(
     U = convert(Vector{T}, StarAlgebras.coeffs(orderunit))
 
     # defining constraints based on the multiplicative structure
-    cnstrs = constraints(parent(elt), augmented=augmented, twisted=true)
+    cnstrs = constraints(parent(elt); augmented = augmented)
 
     prog = ProgressMeter.Progress(
         length(invariant_vectors(wedderburn)),