diff --git a/src/Bridges/Bridges.jl b/src/Bridges/Bridges.jl
index 1e9c0490c5..3e4da6601d 100644
--- a/src/Bridges/Bridges.jl
+++ b/src/Bridges/Bridges.jl
@@ -25,6 +25,7 @@ include("lazybridgeoptimizer.jl")
 
 # This is used by JuMP and removes the need to update JuMP everytime a bridge is added
 MOIU.@model AllBridgedConstraints () (Interval,) (SecondOrderCone, RotatedSecondOrderCone, GeometricMeanCone, LogDetConeTriangle, RootDetConeTriangle) () () (ScalarAffineFunction,) (VectorOfVariables,) (VectorAffineFunction,)
+
 """
     fullbridgeoptimizer(model::MOI.ModelLike, ::Type{T}) where T
 
@@ -33,13 +34,14 @@ Returns a `LazyBridgeOptimizer` bridging `model` for every bridge defined in thi
 function fullbridgeoptimizer(model::MOI.ModelLike, ::Type{T}) where T
     bridgedmodel = MOIB.LazyBridgeOptimizer(model, AllBridgedConstraints{T}())
     addbridge!(bridgedmodel, MOIB.SplitIntervalBridge{T})
+    addbridge!(bridgedmodel, MOIB.RSOCBridge{T})
+    addbridge!(bridgedmodel, MOIB.SOCtoQuadBridge{T})
     addbridge!(bridgedmodel, MOIB.GeoMeanBridge{T})
     addbridge!(bridgedmodel, MOIB.SquarePSDBridge{T})
     addbridge!(bridgedmodel, MOIB.LogDetBridge{T})
     addbridge!(bridgedmodel, MOIB.RootDetBridge{T})
-    addbridge!(bridgedmodel, MOIB.RSOCBridge{T})
-    addbridge!(bridgedmodel, MOIB.RSOCtoPSDCBridge{T})
     addbridge!(bridgedmodel, MOIB.SOCtoPSDCBridge{T})
+    addbridge!(bridgedmodel, MOIB.RSOCtoPSDCBridge{T})
     bridgedmodel
 end
 
@@ -47,6 +49,8 @@ include("intervalbridge.jl")
 @bridge SplitInterval SplitIntervalBridge () (Interval,) () () (SingleVariable,) (ScalarAffineFunction, ScalarQuadraticFunction) () ()
 include("rsocbridge.jl")
 @bridge RSOC RSOCBridge () () (RotatedSecondOrderCone,) () () () (VectorOfVariables,) (VectorAffineFunction,)
+include("soctoquadbridge.jl")
+@bridge SOCtoQuad SOCtoQuadBridge () () (SecondOrderCone,) () () () (VectorOfVariables,) (VectorAffineFunction,)
 include("geomeanbridge.jl")
 @bridge GeoMean GeoMeanBridge () () (GeometricMeanCone,) () () () (VectorOfVariables,) (VectorAffineFunction,)
 include("squarepsdbridge.jl")
diff --git a/src/Bridges/bridge.jl b/src/Bridges/bridge.jl
index 75b237b86f..ff9e389bf0 100644
--- a/src/Bridges/bridge.jl
+++ b/src/Bridges/bridge.jl
@@ -76,3 +76,11 @@ function concrete_bridge_type(bridge_type::DataType,
                               ::Type{<:MOI.AbstractSet})
     return bridge_type
 end
+
+"""
+    need_constraint_primal_fallback(BT::Type{<:AbstractBridge})
+
+Return a `Bool` indicating whether bridges of concrete type `BT` need the bridge
+optimizer to use `get_fallback`.
+"""
+need_constraint_primal_fallback(::Type{<:AbstractBridge}) = false
diff --git a/src/Bridges/bridgeoptimizer.jl b/src/Bridges/bridgeoptimizer.jl
index 25e882dfaa..62c0271f4b 100644
--- a/src/Bridges/bridgeoptimizer.jl
+++ b/src/Bridges/bridgeoptimizer.jl
@@ -208,10 +208,14 @@ end
 # Constraint attributes
 function MOI.get(b::AbstractBridgeOptimizer,
                  attr::MOI.AbstractConstraintAttribute,
-                 ci::CI)
+                 ci::CI{F, S}) where {F, S}
     if isbridged(b, typeof(ci))
         if MOIU.is_result_attribute(attr)
-            MOI.get(b, attr, bridge(b, ci))
+            if attr isa MOI.ConstraintPrimal && need_constraint_primal_fallback(concrete_bridge_type(b, F, S))
+                MOIU.get_fallback(b, attr, ci)
+            else
+                MOI.get(b, attr, bridge(b, ci))
+            end
         else
             MOI.get(b.bridged, attr, ci)
         end
diff --git a/src/Bridges/soctoquadbridge.jl b/src/Bridges/soctoquadbridge.jl
new file mode 100644
index 0000000000..21a6ff4cc6
--- /dev/null
+++ b/src/Bridges/soctoquadbridge.jl
@@ -0,0 +1,83 @@
+"""
+    SOCtoQuadBridge{T}
+
+Constraints of the form `VectorOfVariables`-in-`SecondOrderCone` (resp.
+`VectorAffineFunction`-in-`SecondOrderCone`) can be transformed into a
+`ScalarQuadraticFunction`-in-`GreaterThan` and a
+`SingleVariable`-in-`GreaterThan` (resp.
+`ScalarAffineFunction`-in-`SecondOrderCone`). Indeed, the definition of the
+second-order cone
+```math
+t \\ge || x ||_2 \\}
+```
+is equivalent to
+```math
+\\sum x_i^2 \\le t^2
+```
+with ``t \\ge 0``.
+"""
+struct SOCtoQuadBridge{T, F, G} <: AbstractBridge
+    quad::CI{F, MOI.GreaterThan{T}}
+    t_nonneg::CI{G, MOI.GreaterThan{T}}
+end
+function SOCtoQuadBridge{T, F, G}(model::MOI.ModelLike,
+                                  f::MOI.AbstractVectorFunction,
+                                  s::MOI.SecondOrderCone) where {T, F, G}
+    d = s.dimension
+    f_scalars = MOIU.eachscalar(f)
+    t = f_scalars[1]
+    t_nonneg = MOIU.add_scalar_constraint(model, t, MOI.GreaterThan(zero(T)))
+    quad_f = MOIU.operate(*, T, t, t)
+    for i in 2:d
+        x = f_scalars[i]
+        quad_f = MOIU.operate!(-, T, quad_f, MOIU.operate(*, T, x, x))
+    end
+    quad = MOIU.add_scalar_constraint(model, quad_f, MOI.GreaterThan(zero(T)))
+    return SOCtoQuadBridge{T, F, G}(quad, t_nonneg)
+end
+
+function MOI.supportsconstraint(::Type{SOCtoQuadBridge{T}},
+                                ::Type{<:MOI.AbstractVectorFunction},
+                                ::Type{MOI.SecondOrderCone}) where T
+    return true
+end
+function addedconstrainttypes(::Type{SOCtoQuadBridge{T, F, G}}) where {T, F, G}
+    list = [(F, MOI.GreaterThan{T})]
+    if F != G
+        push!(list, (G, MOI.GreaterThan{T}))
+    end
+    return list
+end
+function concrete_bridge_type(::Type{<:SOCtoQuadBridge{T}},
+                              H::Type{<:MOI.AbstractVectorFunction},
+                              ::Type{MOI.SecondOrderCone}) where T
+    G = MOIU.scalar_type(H)
+    Q = MOIU.promote_operation(*, T, G, G)
+    F = MOIU.promote_operation(-, T, Q, Q)
+    return SOCtoQuadBridge{T, F, G}
+end
+
+# Attributes, Bridge acting as an model
+function MOI.get(b::SOCtoQuadBridge{T, F, G},
+                 ::MOI.NumberOfConstraints{H, MOI.GreaterThan{T}}) where {T, F, G, H}
+    return (F == H) + (G == H)
+end
+function MOI.get(b::SOCtoQuadBridge{T, F, G},
+                 ::MOI.ListOfConstraintIndices{H, MOI.GreaterThan{T}}) where {T, F, G, H}
+    list = CI{H, MOI.GreaterThan{T}}[]
+    if F == H
+        push!(list, b.quad)
+    end
+    if G == H
+        push!(list, b.t_nonneg)
+    end
+    return list
+end
+
+# References
+function MOI.delete!(model::MOI.ModelLike, bridge::SOCtoQuadBridge)
+    MOI.delete!(model, bridge.t_nonneg)
+    MOI.delete!(model, bridge.quad)
+end
+
+need_constraint_primal_fallback(::Type{<:SOCtoQuadBridge}) = true
diff --git a/test/bridge.jl b/test/bridge.jl
index 593bf8e871..8cfd5d7a76 100644
--- a/test/bridge.jl
+++ b/test/bridge.jl
@@ -212,6 +212,34 @@ end
         test_delete_bridge(bridgedmock, ci, 2, ((MOI.VectorAffineFunction{Float64}, MOI.SecondOrderCone, 0),))
     end
 
+    @testset "SOCtoQuad" begin
+        config = MOIT.TestConfig(duals=false)
+        bridgedmock = MOIB.SOCtoQuad{Float64}(mock)
+        mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [1.0, 1/√2, 1/√2])
+        MOIT.soc1vtest(bridgedmock, config)
+        mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [1.0, 1/√2, 1/√2])
+        MOIT.soc1ftest(bridgedmock, config)
+        ci = first(MOI.get(bridgedmock,
+                           MOI.ListOfConstraintIndices{MOI.VectorAffineFunction{Float64},
+                                                       MOI.SecondOrderCone}()))
+        test_delete_bridge(bridgedmock, ci, 3,
+                           ((MOI.ScalarAffineFunction{Float64},
+                             MOI.GreaterThan{Float64}, 0),))
+    end
+
+    @testset "GeoMean" begin
+        mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [ones(4); 2; √2; √2])
+        bridgedmock = MOIB.GeoMean{Float64}(mock)
+        MOIT.geomean1vtest(bridgedmock, config)
+        MOIT.geomean1ftest(bridgedmock, config)
+        # Dual is not yet implemented for GeoMean bridge
+        ci = first(MOI.get(bridgedmock, MOI.ListOfConstraintIndices{MOI.VectorAffineFunction{Float64}, MOI.GeometricMeanCone}()))
+        @test !MOI.supports(bridgedmock, MOI.ConstraintSet(), typeof(ci))
+        @test !MOI.supports(bridgedmock, MOI.ConstraintFunction(), typeof(ci))
+        test_delete_bridge(bridgedmock, ci, 4, ((MOI.VectorAffineFunction{Float64}, MOI.RotatedSecondOrderCone, 0),
+                                                (MOI.ScalarAffineFunction{Float64}, MOI.LessThan{Float64},      1)))
+    end
+
     @testset "SquarePSD" begin
         bridgedmock = MOIB.SquarePSD{Float64}(mock)
         mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, ones(4),
@@ -231,19 +259,31 @@ end
                              MOI.EqualTo{Float64}, 1)))
     end
 
-    @testset "GeoMean" begin
-        mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [ones(4); 2; √2; √2])
-        bridgedmock = MOIB.GeoMean{Float64}(mock)
-        MOIT.geomean1vtest(bridgedmock, config)
-        MOIT.geomean1ftest(bridgedmock, config)
-        # Dual is not yet implemented for GeoMean bridge
-        ci = first(MOI.get(bridgedmock, MOI.ListOfConstraintIndices{MOI.VectorAffineFunction{Float64}, MOI.GeometricMeanCone}()))
+    @testset "LogDet" begin
+        bridgedmock = MOIB.LogDet{Float64}(mock)
+        mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [0, 1, 0, 1, 1, 0, 1, 0, 0])
+        MOIT.logdett1vtest(bridgedmock, config)
+        MOIT.logdett1ftest(bridgedmock, config)
+        # Dual is not yet implemented for LogDet bridge
+        ci = first(MOI.get(bridgedmock, MOI.ListOfConstraintIndices{MOI.VectorAffineFunction{Float64}, MOI.LogDetConeTriangle}()))
         @test !MOI.supports(bridgedmock, MOI.ConstraintSet(), typeof(ci))
         @test !MOI.supports(bridgedmock, MOI.ConstraintFunction(), typeof(ci))
-        test_delete_bridge(bridgedmock, ci, 4, ((MOI.VectorAffineFunction{Float64}, MOI.RotatedSecondOrderCone, 0),
-                                                (MOI.ScalarAffineFunction{Float64}, MOI.LessThan{Float64},      1)))
+        test_delete_bridge(bridgedmock, ci, 4, ((MOI.VectorAffineFunction{Float64}, MOI.ExponentialCone, 0), (MOI.VectorAffineFunction{Float64}, MOI.PositiveSemidefiniteConeTriangle, 0)))
     end
 
+    @testset "RootDet" begin
+        bridgedmock = MOIB.RootDet{Float64}(mock)
+        mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [1, 1, 0, 1, 1, 0, 1])
+        MOIT.rootdett1vtest(bridgedmock, config)
+        MOIT.rootdett1ftest(bridgedmock, config)
+        # Dual is not yet implemented for RootDet bridge
+        ci = first(MOI.get(bridgedmock, MOI.ListOfConstraintIndices{MOI.VectorAffineFunction{Float64}, MOI.RootDetConeTriangle}()))
+        @test !MOI.supports(bridgedmock, MOI.ConstraintSet(), typeof(ci))
+        @test !MOI.supports(bridgedmock, MOI.ConstraintFunction(), typeof(ci))
+        test_delete_bridge(bridgedmock, ci, 4, ((MOI.VectorAffineFunction{Float64}, MOI.RotatedSecondOrderCone, 0),
+                                                (MOI.VectorAffineFunction{Float64}, MOI.GeometricMeanCone, 0),
+                                                (MOI.VectorAffineFunction{Float64}, MOI.PositiveSemidefiniteConeTriangle, 0)))
+    end
     @testset "SOCtoPSD" begin
         bridgedmock = MOIB.SOCtoPSD{Float64}(mock)
         mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [1.0, 1/√2, 1/√2],
@@ -272,29 +312,4 @@ end
         test_delete_bridge(bridgedmock, ci, 2, ((MOI.VectorAffineFunction{Float64}, MOI.PositiveSemidefiniteConeTriangle, 0),))
     end
 
-    @testset "LogDet" begin
-        bridgedmock = MOIB.LogDet{Float64}(mock)
-        mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [0, 1, 0, 1, 1, 0, 1, 0, 0])
-        MOIT.logdett1vtest(bridgedmock, config)
-        MOIT.logdett1ftest(bridgedmock, config)
-        # Dual is not yet implemented for LogDet bridge
-        ci = first(MOI.get(bridgedmock, MOI.ListOfConstraintIndices{MOI.VectorAffineFunction{Float64}, MOI.LogDetConeTriangle}()))
-        @test !MOI.supports(bridgedmock, MOI.ConstraintSet(), typeof(ci))
-        @test !MOI.supports(bridgedmock, MOI.ConstraintFunction(), typeof(ci))
-        test_delete_bridge(bridgedmock, ci, 4, ((MOI.VectorAffineFunction{Float64}, MOI.ExponentialCone, 0), (MOI.VectorAffineFunction{Float64}, MOI.PositiveSemidefiniteConeTriangle, 0)))
-    end
-
-    @testset "RootDet" begin
-        bridgedmock = MOIB.RootDet{Float64}(mock)
-        mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [1, 1, 0, 1, 1, 0, 1])
-        MOIT.rootdett1vtest(bridgedmock, config)
-        MOIT.rootdett1ftest(bridgedmock, config)
-        # Dual is not yet implemented for RootDet bridge
-        ci = first(MOI.get(bridgedmock, MOI.ListOfConstraintIndices{MOI.VectorAffineFunction{Float64}, MOI.RootDetConeTriangle}()))
-        @test !MOI.supports(bridgedmock, MOI.ConstraintSet(), typeof(ci))
-        @test !MOI.supports(bridgedmock, MOI.ConstraintFunction(), typeof(ci))
-        test_delete_bridge(bridgedmock, ci, 4, ((MOI.VectorAffineFunction{Float64}, MOI.RotatedSecondOrderCone, 0),
-                                                (MOI.VectorAffineFunction{Float64}, MOI.GeometricMeanCone, 0),
-                                                (MOI.VectorAffineFunction{Float64}, MOI.PositiveSemidefiniteConeTriangle, 0)))
-    end
 end