diff --git a/src/projection.jl b/src/projection.jl
index ae0c9de73..7a55e956d 100644
--- a/src/projection.jl
+++ b/src/projection.jl
@@ -472,7 +472,6 @@ end
 
 # Triangular
 for UL in (:UpperTriangular, :LowerTriangular, :UpperHessenberg)
-    VERSION < v"1.4" && UL == :UpperHessenberg && continue  # not defined in 1.0
     @eval begin
         ProjectTo(x::$UL) = ProjectTo{$UL}(; data=ProjectTo(parent(x)))
         (project::ProjectTo{$UL})(dx::AbstractArray) = $UL(project.data(dx))
@@ -499,10 +498,10 @@ end
 # Subspaces which aren't subtypes, like Diagonal inside Symmetric above:
 (project::ProjectTo{UpperTriangular})(dx::Diagonal) = project.data(dx)
 (project::ProjectTo{LowerTriangular})(dx::Diagonal) = project.data(dx)
-if VERSION >= v"1.4"
-    (project::ProjectTo{UpperHessenberg})(dx::Diagonal) = project.data(dx)
-    (project::ProjectTo{UpperHessenberg})(dx::UpperTriangular) = project.data(dx)
-end
+
+(project::ProjectTo{UpperHessenberg})(dx::Diagonal) = project.data(dx)
+(project::ProjectTo{UpperHessenberg})(dx::UpperTriangular) = project.data(dx)
+
 (project::ProjectTo{UnitUpperTriangular})(dx::Diagonal) = NoTangent()
 (project::ProjectTo{UnitLowerTriangular})(dx::Diagonal) = NoTangent()
 
diff --git a/test/projection.jl b/test/projection.jl
index 62376dce5..8e7e88be3 100644
--- a/test/projection.jl
+++ b/test/projection.jl
@@ -287,11 +287,9 @@ struct NoSuperType end
         @test pupp(rand(ComplexF32, 3, 3, 1)) isa UpperTriangular{Float64}
         @test ProjectTo(UpperTriangular(randn(3, 3) .> 0))(randn(3, 3)) == NoTangent()
 
-        if VERSION >= v"1.4" # not sure 1.4 exactly!
-            phess = ProjectTo(UpperHessenberg(rand(3, 3)))
-            @test phess(reshape(1:9,3,3)) == [1 4 7; 2 5 8; 0 6 9]
-            @test phess(reshape(1:9,3,3) .+ im) isa UpperHessenberg{Float64}
-        end
+        phess = ProjectTo(UpperHessenberg(rand(3, 3)))
+        @test phess(reshape(1:9,3,3)) == [1 4 7; 2 5 8; 0 6 9]
+        @test phess(reshape(1:9,3,3) .+ im) isa UpperHessenberg{Float64}
 
         pdu = ProjectTo(UnitLowerTriangular(rand(3, 3)))
         # NB, since the diagonal is constant 1, its gradient is zero:
@@ -494,7 +492,7 @@ struct NoSuperType end
         @test eval(Meta.parse(str))(ones(1, 3)) isa Adjoint{Float64,Vector{Float64}}
     end
 
-    VERSION > v"1.1" && @testset "allocation tests" begin
+    @testset "allocation tests" begin
         # For sure these fail on Julia 1.0, not sure about 1.3 etc.
         # We only really care about current stable anyway
         # Each "@test 33 > ..." is zero on nightly, 32 on 1.5.