diff --git a/src/diagonal.jl b/src/diagonal.jl
index 9ce9783c8..498d10d2f 100644
--- a/src/diagonal.jl
+++ b/src/diagonal.jl
@@ -342,7 +342,12 @@ function (*)(D::Diagonal, V::AbstractVector)
     return D.diag .* V
 end
 
+_wrapperop_reverses_products(::Adjoint, ::Diagonal) = true
+_wrapperop_reverses_products(::Transpose{<:Union{Real,Complex}}, ::Diagonal{<:Union{Real,Complex}}) = true
+_wrapperop_reverses_products(A, D) = false
+
 function mul(A::AdjOrTransAbsMat, D::Diagonal)
+    _wrapperop_reverses_products(A, D) || return @invoke mul(A::AbstractMatrix, D::AbstractMatrix)
     adj = _wrapperop(A)
     copy(adj(adj(D) * adj(A)))
 end
@@ -350,6 +355,7 @@ function mul(A::AdjOrTransAbsMat{<:Number, <:StridedMatrix}, D::Diagonal{<:Numbe
     @invoke mul(A::AbstractMatrix, D::AbstractMatrix)
 end
 function mul(D::Diagonal, A::AdjOrTransAbsMat)
+    _wrapperop_reverses_products(A, D) || return @invoke mul(D::AbstractMatrix, A::AbstractMatrix)
     adj = _wrapperop(A)
     copy(adj(adj(A) * adj(D)))
 end
@@ -370,6 +376,7 @@ end
 # A' = A' * D => A = D' * A
 # This uses the fact that D' is a Diagonal
 function rmul!(A::AdjOrTransAbsMat, D::Diagonal)
+    _wrapperop_reverses_products(A, D) || return @invoke rmul!(A::AbstractMatrix, D::Diagonal)
     f = _wrapperop(A)
     lmul!(f(D), f(A))
     A
@@ -420,6 +427,7 @@ end
 # A' = D * A' => A = A * D'
 # This uses the fact that D' is a Diagonal
 function lmul!(D::Diagonal, A::AdjOrTransAbsMat)
+    _wrapperop_reverses_products(A, D) || return @invoke lmul!(D::Diagonal, A::AbstractVecOrMat)
     f = _wrapperop(A)
     rmul!(f(A), f(D))
     A
diff --git a/src/triangular.jl b/src/triangular.jl
index 93fa0fcdc..a97525f3d 100644
--- a/src/triangular.jl
+++ b/src/triangular.jl
@@ -1841,7 +1841,7 @@ function generic_trimatdiv!(C::AbstractVecOrMat, uploc, isunitc, ::Function, xA:
                 for i in axes(B,1)[2:end]
                     C[i,k] = oA \ B[i,k] - _ustrip(conj(A[i,1])) * C1
                 end
-                for j in axes(A,2)
+                for j in axes(A,2)[2:end]
                     Cj = C[j,k]
                     for i in j+1:lastindex(A,1)
                         C[i,k] -= _ustrip(conj(A[i,j])) * Cj
diff --git a/test/diagonal.jl b/test/diagonal.jl
index ef438037a..1793c63d5 100644
--- a/test/diagonal.jl
+++ b/test/diagonal.jl
@@ -16,6 +16,7 @@ using Main.LinearAlgebraTestHelpers.InfiniteArrays
 using Main.LinearAlgebraTestHelpers.FillArrays
 using Main.LinearAlgebraTestHelpers.SizedArrays
 using Main.LinearAlgebraTestHelpers.ImmutableArrays
+using Main.LinearAlgebraTestHelpers.Quaternions
 
 const n=12 # Size of matrix problem to test
 Random.seed!(1)
@@ -1578,4 +1579,17 @@ end
     @test D == D2
 end
 
+@testset "products of Diagonal with transposed non-commutative matrices (issue #1750)" begin
+    i, j, k = Quaternion(0.0,1,0,0), Quaternion(0.0,0,1,0), Quaternion(0.0,0,0,1)
+    M = [i j; zero(i) k]
+    A = view(M, [1, 2], :)
+    D = Diagonal([j, k])
+    for op in (transpose, adjoint)
+        @test op(A) * D == Matrix(op(A)) * D
+        @test D * op(A) == D * Matrix(op(A))
+        @test rmul!(op(copy(M)), D) == Matrix(op(M)) * D
+        @test lmul!(D, op(copy(M))) == D * Matrix(op(M))
+    end
+end
+
 end # module TestDiagonal
diff --git a/test/triangular.jl b/test/triangular.jl
index e24f62ab6..65abf42d8 100644
--- a/test/triangular.jl
+++ b/test/triangular.jl
@@ -1118,4 +1118,17 @@ end
     end
 end
 
+@testset "issue #1749: ldiv! with lazy adjoint/transpose parent" begin
+    M = ComplexF64[1 2+3im 1im; 4-1im 5 2-im; 3+im 2im 7]
+    b = ComplexF64[1, 2, 3]
+    B = ComplexF64[1 4; 2 5; 3 6]
+    for P in (transpose(M'), adjoint(transpose(M)), transpose(M), M')
+        for W in (UpperTriangular, UnitUpperTriangular, LowerTriangular, UnitLowerTriangular)
+            A = W(P)
+            @test A \ b ≈ Matrix(A) \ b ≈ lu(Matrix(A)) \ b
+            @test A \ B ≈ Matrix(A) \ B ≈ lu(Matrix(A)) \ B
+        end
+    end
+end
+
 end # module TestTriangular