diff --git a/src/PropertyT.jl b/src/PropertyT.jl
index 60b0aef..f5f3861 100644
--- a/src/PropertyT.jl
+++ b/src/PropertyT.jl
@@ -25,6 +25,8 @@ include("gradings.jl")
 
 include("actions/actions.jl")
 
+include("orbit_basis.jl")
+
 function group_algebra(G::Groups.Group, S = gens(G); halfradius::Integer)
     S = union!(S, inv.(S))
     @info "generating wl-metric ball of radius $(2halfradius)"
diff --git a/src/orbit_basis.jl b/src/orbit_basis.jl
new file mode 100644
index 0000000..8fa3ea0
--- /dev/null
+++ b/src/orbit_basis.jl
@@ -0,0 +1,360 @@
+function minimal_in_orbit(f, x, group, act::SymbolicWedderburn.Action)
+    stab_size = UInt32(0)
+    min, min_elt = f(x), x
+    for g in group
+        y = action(act, g, x)
+        if y == x
+            stab_size += 1
+            continue
+        else
+            fy = f(y)
+            if fy < min
+                min, min_elt = fy, y
+            end
+        end
+    end
+
+    return min_elt, stab_size
+end
+
+function saturate!(
+    f,
+    gens,
+    elements,
+    stab_sizes,
+    hashes = Set(elements);
+    start_at = firstindex(elements),
+)
+    g_shifted = similar(gens)
+    s_sizes = similar(gens, UInt32)
+    already_seen = similar(gens, Bool)
+    range = start_at:lastindex(elements)
+    ProgressMeter.@showprogress 2.0 "Computing Orbit representatives: $(length(range))" for idx in
+                                                                                            range
+        g = elements[idx]
+
+        Threads.@threads for i in eachindex(gens)
+            h, s_size = f(g * gens[i])
+            g_shifted[i] = h
+            s_sizes[i] = s_size
+            already_seen[i] = h in hashes
+        end
+        # t = g_shifted[.!seen]
+        # t = unique!(t)
+        # append!(old, t)
+        # union!(hashes, t)
+        for (seen, g, ssize) in zip(already_seen, g_shifted, s_sizes)
+            if !seen
+                if !(g in hashes)
+                    push!(elements, g)
+                    push!(stab_sizes, ssize)
+                    push!(hashes, g)
+                end
+            end
+        end
+    end
+    return elements, stab_sizes, hashes
+end
+
+function orbit_reps(
+    seed_elts,
+    Σ::Groups.Group,
+    act::SymbolicWedderburn.Action;
+    radius::Integer,
+    minimize = hash,
+)
+    @assert !isempty(seed_elts)
+    @assert radius ≥ 0
+
+    if Groups.order(Int, Σ) < 2^15
+        f = x -> minimal_in_orbit(minimize, x, collect(Σ), act)
+    else
+        f = x -> minimal_in_orbit(minimize, x, Σ, act)
+    end
+
+    orbits = [one(first(seed_elts))]
+    stab_sizes = [order(UInt32, Σ)]
+    hashes = Set(orbits)
+    sizes = ones(Int, radius + 1)
+
+    start_at = sizes[1]
+    for r in 1:radius
+        orbits, hashes = saturate!(
+            f,
+            seed_elts,
+            orbits,
+            stab_sizes,
+            hashes;
+            start_at = start_at,
+        )
+        sizes[r+1] = length(orbits)
+        start_at = sizes[r]
+    end
+
+    return (orbits, stab_sizes), sizes[begin+1:end]
+end
+
+###
+# move to StarAlgebras
+
+import SparseArrays
+function SparseArrays.indtype(b::StarAlgebras.AbstractBasis)
+    return SparseArrays.indtype(typeof(b))
+end
+function SparseArrays.indtype(
+    ::Type{<:StarAlgebras.AbstractBasis{T,I}},
+) where {T,I}
+    return I
+end
+
+###
+
+struct OrbitBasis{T,I,G,A,F} <: StarAlgebras.AbstractBasis{T,I}
+    orbit_reps::StarAlgebras.Basis{T,I}
+    stab_size::Vector{UInt32}
+    group::G
+    action::A
+    __minimize_function::F
+
+    function OrbitBasis(
+        orbit_reps::StarAlgebras.AbstractBasis{T,I},
+        grp,
+        act::SymbolicWedderburn.Action,
+        minimize,
+        stab_size::AbstractVector{<:Integer},
+    ) where {T,I}
+        return new{T,I,typeof(grp),typeof(act),typeof(minimize)}(
+            orbits_reps,
+            grp,
+            act,
+            stab_size,
+            minimize,
+        )
+    end
+end
+
+__group_of(ob::OrbitBasis) = ob.group
+SymbolicWedderburn.action(ob::OrbitBasis) = ob.action
+
+Base.size(ob::OrbitBasis) = size(ob.orbit_reps)
+Base.getindex(ob::OrbitBasis, i::Integer) = ob.orbit_reps[i]
+function Base.getindex(ob::OrbitBasis{T}, x::T) where {T}
+    if x in ob.orbit_reps
+        return ob.orbit_reps[x]
+    else
+        for σ in __group_of(ob)
+            y = action(action(ob), σ, x)
+            y in ob.orbit_reps && return ob.orbit_reps[y]
+        end
+        argmin(ob.__minimize_function)
+        throw(KeyError(minimal_in_orbit(hash, x, __group_of(ob), action(ob))))
+        # y = minimal_in_orbit(hash, x, __group_of(ob), action(ob))
+        # y in ob.basis ? return ob.basis[y] : throw("NotInBasis")
+    end
+end
+
+function _mtable(
+    basis::StarAlgebras.AbstractBasis{T},
+    orbits::OrbitBasis{T},
+    table_size,
+) where {T}
+    M = zeros(SparseArrays.indtype(orbits), table_size)
+    starof = zeros(eltype(M), size(M, 1))
+
+    # it is safe to do Threads.@threads here, since
+    # in case of race condition (two threads overwriting a single entry in M)
+    # * M[idx, jdx] is 0 at the beginning,
+    # * M[idx, jdx] will be written by two threads
+    # thread, we will be computing and overwriting with the same value;
+    Threads.@threads for jdx in axes(M, 2)
+        b = basis[jdx]
+        starof[jdx] = basis[StarAlgebras.star(b)]
+        for idx in axes(M, 1)
+            !iszero(M[idx, jdx]) && continue
+            a = basis[idx]
+            k = orbits[a*b]
+            for g in orbits.group
+                aᵍ = action(action(orbits), g, a)
+                bᵍ = action(action(orbits), g, b)
+                M[basis[aᵍ], basis[bᵍ]] = k
+            end
+        end
+    end
+
+    @assert all(!iszero, starof)
+    @assert all(!iszero, M)
+    return StarAlgebras.MTable(M, starof)
+end
+
+function constraints(
+    basis_half::StarAlgebras.AbstractBasis,
+    orbits::OrbitBasis;
+    augmented::Bool,
+    basis_support,
+)
+    l = length(basis_support)
+    mstr = _mtable(basis_half, orbits, (l, l))
+    id = orbits[one(first(basis_half))]
+    cnstrs = _constraints(mstr; augmented = augmented, id = id)
+
+    ordG = length(__group_of(orbits))
+
+    return Dict(
+        orbits[i] =>
+            ConstraintMatrix(c, size(mstr)..., orbits.stab_size[i] / ordG) for
+        (i, c) in pairs(cnstrs) if !isempty(c)
+    )
+end
+
+## move to Tests
+#=
+mt = let R = 2, sizes = sizes
+    mt = @time _mtable(basis(RG), obasis, (sizes[R], sizes[R]))
+    # k = findlast(o -> basis(RG)[o] ≤ sizes[R], orbits)
+    # @assert k == osizes[2R]
+    @assert all(≤(osizes[2R]), mt)
+    @assert all(>(0), mt)
+
+    for i in 1:sizes[1]
+        for j in 1:sizes[1]
+            @assert obasis[basis(RG)[i]*basis(RG)[j]] == mt[i, j]
+        end
+    end
+
+    @assert maximum(mt) == osizes[2R]
+
+    C = PropertyT._constraints(mt; augmented = true, id = 1)
+    @assert length(C) == osizes[2R]
+    @assert all(!isempty, C)
+    mt
+end
+=#
+
+function SymbolicWedderburn.WedderburnDecomposition(
+    T::Type,
+    G::Groups.Group,
+    action::SymbolicWedderburn.Action,
+    orbit_basis::OrbitBasis,
+    basis_half,
+    S = Rational{Int};
+    semisimple = false,
+)
+    tbl = SymbolicWedderburn.CharacterTable(S, G)
+
+    ehom = SymbolicWedderburn.CachedExtensionHomomorphism(
+        G,
+        action,
+        basis_half;
+        precompute = true,
+    )
+
+    Uπs = SymbolicWedderburn.symmetry_adapted_basis(
+        T,
+        tbl,
+        ehom;
+        semisimple = semisimple,
+    )
+    return SymbolicWedderburn.WedderburnDecomposition(
+        basis_half,
+        orbit_basis,
+        Uπs,
+        ehom,
+    )
+end
+
+function sos_problem_primal(
+    elt::StarAlgebras.AlgebraElement,
+    order_unit::StarAlgebras.AlgebraElement,
+    wedderburn::WedderburnDecomposition{B,<:OrbitBasis};
+    upper_bound = Inf,
+    augmented::Bool = iszero(StarAlgebras.aug(elt)) &&
+        iszero(StarAlgebras.aug(order_unit)),
+    check_orthogonality = true,
+    show_progress = false,
+    constraints = PropertyT.constraints(
+        basis(wedderburn),
+        invariant_vectors(wedderburn);
+        augmented = augmented,
+        basis_support = 1:length(basis(wedderburn)),
+    ),
+) where {B}
+    @assert parent(elt) === parent(order_unit)
+    if check_orthogonality
+        if any(!isorth_projection, direct_summands(wedderburn))
+            error(
+                "Wedderburn decomposition contains a non-orthogonal projection",
+            )
+        end
+    end
+
+    prog = ProgressMeter.Progress(
+        length(invariant_vectors(wedderburn));
+        dt = 1,
+        desc = "Adding constraints: ",
+        enabled = show_progress,
+        barlen = 60,
+        showspeed = true,
+    )
+
+    feasibility_problem = iszero(order_unit)
+
+    # problem creation starts here
+    model = JuMP.Model()
+    if !feasibility_problem # add λ of not?
+        JuMP.@variable(model, λ)
+        JuMP.@objective(model, Max, λ)
+    end
+    if isfinite(upper_bound)
+        if feasibility_problem
+            @warn "setting `upper_bound` with zero `unit` has no effect"
+        else
+            JuMP.@constraint(model, λ <= upper_bound)
+        end
+    end
+
+    # semidefinite constraints as described by wedderburn
+    Ps = map(direct_summands(wedderburn)) do ds
+        dim = size(ds, 1)
+        P = JuMP.@variable(model, [1:dim, 1:dim], Symmetric)
+        JuMP.@constraint(model, P in PSDCone())
+        return P
+    end
+
+    begin # Ms are preallocated for the constraints loop
+        T = eltype(wedderburn)
+        Ms = [spzeros.(T, size(p)...) for p in Ps]
+        _eps = 10 * eps(T) * max(size(parent(elt).mstructure)...)
+    end
+
+    id_one = findfirst(isone, invariant_vectors(wedderburn))
+    # adding linear constraints: one per orbit
+    for (i, r) in enumerate(invariant_vectors(wedderburn))
+        ProgressMeter.next!(prog; showvalues = __show_itrs(i, prog.n))
+        augmented && i == id_one && continue
+        A_r = sparse(constraints[r])
+
+        Ms = SymbolicWedderburn.diagonalize!(
+            Ms,
+            A_r,
+            wedderburn;
+            trace_preserving = true,
+        )
+        for M in Ms
+            SparseArrays.droptol!(M, _eps)
+        end
+        if r in basis(parent(elt))
+            if feasibility_problem
+                JuMP.@constraint(model, elt(r) == _dot(Ps, Ms))
+            else
+                JuMP.@constraint(
+                    model,
+                    elt(r) - λ * order_unit(r) == _dot(Ps, Ms)
+                )
+            end
+        else
+            JuMP.@constraint(model, 0 == PropertyT._dot(Ps, Ms))
+        end
+    end
+    ProgressMeter.finish!(prog)
+    return model, Ps
+end