@testset "1712.07167 Examples" begin
    @testset "SAut(F₃)" begin
        N = 3
        G = SpecialAutomorphismGroup(FreeGroup(N))
        @info "running tests for" G
        RG, S, sizes = PropertyT.group_algebra(G; halfradius = 2)

        P = PG.PermGroup(PG.perm"(1,2)", PG.Perm(circshift(1:N, -1)))
        Σ = Groups.Constructions.WreathProduct(PG.PermGroup(PG.perm"(1,2)"), P)
        act = PropertyT.action_by_conjugation(G, Σ)
        wd = SW.WedderburnDecomposition(
            Float64,
            Σ,
            act,
            SA.basis(RG),
            SA.Basis{UInt16}(@view SA.basis(RG)[1:sizes[2]]),
        )
        @info wd

        Δ = let RG = RG, S = S
            RG(length(S)) - sum(RG(s) for s in S)
        end

        elt = Δ^2
        unit = Δ
        ub = Inf

        status, certified, λ_cert = check_positivity(
            elt,
            unit,
            wd;
            upper_bound = ub,
            halfradius = 2,
            optimizer = cosmo_optimizer(;
                eps = 1e-7,
                max_iters = 10_000,
                accel = 50,
                alpha = 1.9,
            ),
        )

        @test status == JuMP.OPTIMAL
        @test !certified
        @test λ_cert < 0
    end

    @testset "SL(3,Z) has (T)" begin
        n = 3

        SL = MatrixGroups.SpecialLinearGroup{n}(Int8)
        @info "running tests for" SL
        RSL, S, sizes = PropertyT.group_algebra(SL; halfradius = 2)

        Δ = RSL(length(S)) - sum(RSL(s) for s in S)

        @testset "Wedderburn formulation" begin
            P = PG.PermGroup(PG.perm"(1,2)", PG.Perm(circshift(1:n, -1)))
            Σ = Groups.Constructions.WreathProduct(
                PG.PermGroup(PG.perm"(1,2)"),
                P,
            )
            act = PropertyT.action_by_conjugation(SL, Σ)
            wd = SW.WedderburnDecomposition(
                Rational{Int},
                Σ,
                act,
                SA.basis(RSL),
                SA.Basis{UInt16}(@view SA.basis(RSL)[1:sizes[2]]),
            )
            @info wd

            elt = Δ^2
            unit = Δ
            ub = 0.2801

            @test_throws ErrorException PropertyT.sos_problem_primal(
                elt,
                unit,
                wd,
                upper_bound = ub,
                augmented = false,
            )

            wdfl = SW.WedderburnDecomposition(
                Float64,
                Σ,
                act,
                SA.basis(RSL),
                SA.Basis{UInt16}(@view SA.basis(RSL)[1:sizes[2]]),
            )

            model, varP = PropertyT.sos_problem_primal(
                elt,
                unit,
                wdfl;
                upper_bound = ub,
                augmented = false,
            )

            status, warm = PropertyT.solve(
                model,
                cosmo_optimizer(;
                    eps = 1e-10,
                    max_iters = 20_000,
                    accel = 50,
                    alpha = 1.9,
                ),
            )

            @test status == JuMP.OPTIMAL

            status, _ = PropertyT.solve(
                model,
                scs_optimizer(;
                    eps = 1e-10,
                    max_iters = 100,
                    accel = -20,
                    alpha = 1.2,
                ),
                warm,
            )

            @test status == JuMP.OPTIMAL

            Q = @time let varP = varP
                Qs = map(varP) do P
                    return real.(sqrt(JuMP.value.(P)))
                end
                PropertyT.reconstruct(Qs, wdfl)
            end
            λ = JuMP.value(model[:λ])

            sos = PropertyT.compute_sos(parent(elt), Q; augmented = false)

            certified, λ_cert = PropertyT.certify_solution(
                elt,
                unit,
                λ,
                Q;
                halfradius = 2,
                augmented = false,
            )

            @test certified
            @test λ_cert >= 28 // 100
        end

        @testset "augmented Wedderburn formulation" begin
            elt = Δ^2
            unit = Δ
            ub = Inf

            P = PG.PermGroup(PG.perm"(1,2)", PG.Perm(circshift(1:n, -1)))
            Σ = Groups.Constructions.WreathProduct(
                PG.PermGroup(PG.perm"(1,2)"),
                P,
            )
            act = PropertyT.action_by_conjugation(SL, Σ)

            wdfl = SW.WedderburnDecomposition(
                Float64,
                Σ,
                act,
                SA.basis(RSL),
                SA.Basis{UInt16}(@view SA.basis(RSL)[1:sizes[2]]),
            )
            @info wdfl

            opt_problem, varP = PropertyT.sos_problem_primal(
                elt,
                unit,
                wdfl;
                upper_bound = ub,
                # augmented = true # since both elt and unit are augmented
            )

            status, _ = PropertyT.solve(
                opt_problem,
                cosmo_optimizer(;
                    eps = 1e-8,
                    max_iters = 20_000,
                    accel = 50,
                    alpha = 1.9,
                ),
            )

            @test status == JuMP.OPTIMAL

            Q = @time let varP = varP
                Qs = map(varP) do P
                    return real.(sqrt(JuMP.value.(P)))
                end
                PropertyT.reconstruct(Qs, wdfl)
            end

            certified, λ_cert = PropertyT.certify_solution(
                elt,
                unit,
                JuMP.objective_value(opt_problem),
                Q;
                halfradius = 2,
                # augmented = true # since both elt and unit are augmented
            )

            @test certified
            @test λ_cert > 28 // 100
        end
    end
end