diff --git a/Project.toml b/Project.toml
index d1c03f7..dbbe6bb 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "KroneckerArrays"
 uuid = "05d0b138-81bc-4ff7-84be-08becefb1ccc"
 authors = ["ITensor developers <support@itensor.org> and contributors"]
-version = "0.1.21"
+version = "0.1.22"
 
 [deps]
 Adapt = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
@@ -23,7 +23,7 @@ KroneckerArraysBlockSparseArraysExt = ["BlockArrays", "BlockSparseArrays"]
 [compat]
 Adapt = "4.3.0"
 BlockArrays = "1.6"
-BlockSparseArrays = "0.7.21"
+BlockSparseArrays = "0.7.22"
 DerivableInterfaces = "0.5.0"
 DiagonalArrays = "0.3.5"
 FillArrays = "1.13.0"
diff --git a/ext/KroneckerArraysBlockSparseArraysExt/KroneckerArraysBlockSparseArraysExt.jl b/ext/KroneckerArraysBlockSparseArraysExt/KroneckerArraysBlockSparseArraysExt.jl
index 9fa1018..b23c869 100644
--- a/ext/KroneckerArraysBlockSparseArraysExt/KroneckerArraysBlockSparseArraysExt.jl
+++ b/ext/KroneckerArraysBlockSparseArraysExt/KroneckerArraysBlockSparseArraysExt.jl
@@ -99,4 +99,10 @@ function (f::GetUnstoredBlock)(
   return error("Not implemented.")
 end
 
+using BlockSparseArrays: BlockSparseArrays
+using KroneckerArrays: KroneckerArrays, KroneckerVector
+function BlockSparseArrays.to_truncated_indices(values::KroneckerVector, I)
+  return KroneckerArrays.to_truncated_indices(values, I)
+end
+
 end
diff --git a/src/cartesianproduct.jl b/src/cartesianproduct.jl
index e3eb2fe..e0138f3 100644
--- a/src/cartesianproduct.jl
+++ b/src/cartesianproduct.jl
@@ -26,10 +26,16 @@ arguments(a::CartesianProduct, n::Int) = arguments(a)[n]
 arg1(a::CartesianProduct) = a.a
 arg2(a::CartesianProduct) = a.b
 
+Base.copy(a::CartesianProduct) = copy(arg1(a)) × copy(arg2(a))
+
 function Base.show(io::IO, a::CartesianProduct)
   print(io, a.a, " × ", a.b)
   return nothing
 end
+function Base.show(io::IO, ::MIME"text/plain", a::CartesianProduct)
+  show(io, a)
+  return nothing
+end
 
 ×(a::AbstractVector, b::AbstractVector) = CartesianProduct(a, b)
 Base.length(a::CartesianProduct) = length(a.a) * length(a.b)
@@ -42,8 +48,38 @@ function Base.getindex(a::CartesianProduct, i::CartesianPair)
   return arg1(a)[arg1(i)] × arg2(a)[arg2(i)]
 end
 function Base.getindex(a::CartesianProduct, i::Int)
-  I = Tuple(CartesianIndices((length(arg1(a)), length(arg2(a))))[i])
-  return a[I[1] × I[2]]
+  I = Tuple(CartesianIndices((length(arg2(a)), length(arg1(a))))[i])
+  return a[I[2] × I[1]]
+end
+
+struct CartesianProductVector{T,P<:CartesianProduct,V<:AbstractVector{T}} <:
+       AbstractVector{T}
+  product::P
+  values::V
+end
+cartesianproduct(r::CartesianProductVector) = getfield(r, :product)
+unproduct(r::CartesianProductVector) = getfield(r, :values)
+Base.length(a::CartesianProductVector) = length(unproduct(a))
+Base.size(a::CartesianProductVector) = (length(a),)
+function Base.axes(r::CartesianProductVector)
+  return (CartesianProductUnitRange(cartesianproduct(r), only(axes(unproduct(r)))),)
+end
+function Base.copy(a::CartesianProductVector)
+  return CartesianProductVector(copy(cartesianproduct(a)), copy(unproduct(a)))
+end
+function Base.getindex(r::CartesianProductVector, i::Integer)
+  return unproduct(r)[i]
+end
+
+function Base.show(io::IO, a::CartesianProductVector)
+  show(io, unproduct(a))
+  return nothing
+end
+function Base.show(io::IO, mime::MIME"text/plain", a::CartesianProductVector)
+  show(io, mime, cartesianproduct(a))
+  println(io)
+  show(io, mime, unproduct(a))
+  return nothing
 end
 
 struct CartesianProductUnitRange{T,P<:CartesianProduct,R<:AbstractUnitRange{T}} <:
@@ -60,13 +96,24 @@ unproduct(r::CartesianProductUnitRange) = getfield(r, :range)
 arg1(a::CartesianProductUnitRange) = arg1(cartesianproduct(a))
 arg2(a::CartesianProductUnitRange) = arg2(cartesianproduct(a))
 
+function Base.show(io::IO, a::CartesianProductUnitRange)
+  show(io, unproduct(a))
+  return nothing
+end
+function Base.show(io::IO, mime::MIME"text/plain", a::CartesianProductUnitRange)
+  show(io, mime, cartesianproduct(a))
+  println(io)
+  show(io, mime, unproduct(a))
+  return nothing
+end
+
 function CartesianProductUnitRange(p::CartesianProduct)
   return CartesianProductUnitRange(p, Base.OneTo(length(p)))
 end
 function CartesianProductUnitRange(a, b)
   return CartesianProductUnitRange(a × b)
 end
-to_product_indices(a::AbstractUnitRange) = a
+to_product_indices(a::AbstractVector) = a
 to_product_indices(i::Integer) = Base.OneTo(i)
 cartesianrange(a, b) = cartesianrange(to_product_indices(a) × to_product_indices(b))
 function cartesianrange(p::CartesianPair)
@@ -94,10 +141,16 @@ function Base.checkindex(::Type{Bool}, inds::CartesianProductUnitRange, i::Carte
   return checkindex(Bool, arg1(inds), arg1(i)) && checkindex(Bool, arg2(inds), arg2(i))
 end
 
+function Base.getindex(a::CartesianProductUnitRange, I::CartesianProduct)
+  prod = cartesianproduct(a)
+  prod_I = arg1(prod)[arg1(I)] × arg2(prod)[arg2(I)]
+  return CartesianProductVector(prod_I, map(Base.Fix1(getindex, a), I))
+end
+
 # Reverse map from CartesianPair to linear index in the range.
 function Base.getindex(inds::CartesianProductUnitRange, i::CartesianPair)
-  i′ = (findfirst(==(arg1(i)), arg1(inds)), findfirst(==(arg2(i)), arg2(inds)))
-  return inds[LinearIndices((length(arg1(inds)), length(arg2(inds))))[i′...]]
+  i′ = (findfirst(==(arg2(i)), arg2(inds)), findfirst(==(arg1(i)), arg1(inds)))
+  return inds[LinearIndices((length(arg2(inds)), length(arg1(inds))))[i′...]]
 end
 
 using Base.Broadcast: DefaultArrayStyle
diff --git a/src/fillarrays/kroneckerarray.jl b/src/fillarrays/kroneckerarray.jl
index fc40ef7..f943ce0 100644
--- a/src/fillarrays/kroneckerarray.jl
+++ b/src/fillarrays/kroneckerarray.jl
@@ -22,7 +22,13 @@ const KroneckerSquareEye{T,A<:AbstractMatrix{T},B<:SquareEye{T}} = KroneckerMatr
 const SquareEyeSquareEye{T,A<:SquareEye{T},B<:SquareEye{T}} = KroneckerMatrix{T,A,B}
 
 _getindex(a::Eye, I1::Colon, I2::Colon) = a
+_getindex(a::Eye, I1::Base.Slice, I2::Base.Slice) = a
+_getindex(a::Eye, I1::Base.Slice, I2::Colon) = a
+_getindex(a::Eye, I1::Colon, I2::Base.Slice) = a
 _view(a::Eye, I1::Colon, I2::Colon) = a
+_view(a::Eye, I1::Base.Slice, I2::Base.Slice) = a
+_view(a::Eye, I1::Base.Slice, I2::Colon) = a
+_view(a::Eye, I1::Colon, I2::Base.Slice) = a
 
 # Like `adapt` but preserves `Eye`.
 _adapt(to, a::Eye) = a
diff --git a/src/fillarrays/matrixalgebrakit_truncate.jl b/src/fillarrays/matrixalgebrakit_truncate.jl
index e505cf2..1bf6ed2 100644
--- a/src/fillarrays/matrixalgebrakit_truncate.jl
+++ b/src/fillarrays/matrixalgebrakit_truncate.jl
@@ -20,34 +20,40 @@ const OnesKroneckerVector{T,A<:OnesVector{T},B<:AbstractVector{T}} = KroneckerVe
 const KroneckerOnesVector{T,A<:AbstractVector{T},B<:OnesVector{T}} = KroneckerVector{T,A,B}
 const OnesVectorOnesVector{T,A<:OnesVector{T},B<:OnesVector{T}} = KroneckerVector{T,A,B}
 
-function MatrixAlgebraKit.findtruncated(
-  values::OnesKroneckerVector, strategy::KroneckerTruncationStrategy
-)
-  I = findtruncated(Vector(values), strategy.strategy)
-  prods = collect(cartesianproduct(only(axes(values))))[I]
-  I_data = unique(map(arg1, prods))
+axis(a) = only(axes(a))
+
+# Convert indices determined with a generic call to `findtruncated` to indices
+# more suited for a KroneckerVector.
+function to_truncated_indices(values::OnesKroneckerVector, I)
+  prods = cartesianproduct(axis(values))[I]
+  I_id = only(to_indices(arg1(values), (:,)))
+  I_data = unique(arg2.(prods))
   # Drop truncations that occur within the identity.
   I_data = filter(I_data) do i
-    return count(x -> arg1(x) == i, prods) == length(arg1(values))
+    return count(x -> arg2(x) == i, prods) == length(arg2(values))
   end
-  return (:) × I_data
+  return I_id × I_data
 end
-function MatrixAlgebraKit.findtruncated(
-  values::KroneckerOnesVector, strategy::KroneckerTruncationStrategy
-)
-  I = findtruncated(Vector(values), strategy.strategy)
-  prods = collect(cartesianproduct(only(axes(values))))[I]
-  I_data = unique(map(x -> arg2(x), prods))
+function to_truncated_indices(values::KroneckerOnesVector, I)
+  #I = findtruncated(Vector(values), strategy.strategy)
+  prods = cartesianproduct(axis(values))[I]
+  I_data = unique(arg1.(prods))
   # Drop truncations that occur within the identity.
   I_data = filter(I_data) do i
-    return count(x -> arg2(x) == i, prods) == length(arg2(values))
+    return count(x -> arg1(x) == i, prods) == length(arg2(values))
   end
-  return I_data × (:)
+  I_id = only(to_indices(arg2(values), (:,)))
+  return I_data × I_id
+end
+function to_truncated_indices(values::OnesVectorOnesVector, I)
+  return throw(ArgumentError("Can't truncate Eye ⊗ Eye."))
 end
+
 function MatrixAlgebraKit.findtruncated(
-  values::OnesVectorOnesVector, strategy::KroneckerTruncationStrategy
+  values::KroneckerVector, strategy::KroneckerTruncationStrategy
 )
-  return throw(ArgumentError("Can't truncate Eye ⊗ Eye."))
+  I = findtruncated(Vector(values), strategy.strategy)
+  return to_truncated_indices(values, I)
 end
 
 for f in [:eig_trunc!, :eigh_trunc!]
diff --git a/src/kroneckerarray.jl b/src/kroneckerarray.jl
index 94d601a..96167f3 100644
--- a/src/kroneckerarray.jl
+++ b/src/kroneckerarray.jl
@@ -167,13 +167,22 @@ function Base.getindex(a::KroneckerArray{<:Any,N}, I::Vararg{Integer,N}) where {
   return a[I′...]
 end
 
+# Indexing logic.
+function Base.to_indices(
+  a::KroneckerArray, inds, I::Tuple{Union{CartesianPair,CartesianProduct},Vararg}
+)
+  I1 = to_indices(arg1(a), arg1.(inds), arg1.(I))
+  I2 = to_indices(arg2(a), arg2.(inds), arg2.(I))
+  return I1 .× I2
+end
+
 # Allow customizing for `FillArrays.Eye`.
 _getindex(a::AbstractArray, I...) = a[I...]
-function Base.getindex(a::KroneckerArray{<:Any,N}, I::Vararg{CartesianProduct,N}) where {N}
-  return _getindex(arg1(a), arg1.(I)...) ⊗ _getindex(arg2(a), arg2.(I)...)
-end
-function Base.getindex(a::KroneckerArray{<:Any,N}, I::Vararg{CartesianPair,N}) where {N}
-  return _getindex(arg1(a), arg1.(I)...) ⊗ _getindex(arg2(a), arg2.(I)...)
+function Base.getindex(
+  a::KroneckerArray{<:Any,N}, I::Vararg{Union{CartesianPair,CartesianProduct},N}
+) where {N}
+  I′ = to_indices(a, I)
+  return _getindex(arg1(a), arg1.(I′)...) ⊗ _getindex(arg2(a), arg2.(I′)...)
 end
 # Fix ambigiuity error.
 Base.getindex(a::KroneckerArray{<:Any,0}) = arg1(a)[] * arg2(a)[]
diff --git a/test/test_basics.jl b/test/test_basics.jl
index 0fe2ef3..ee1ec6c 100644
--- a/test/test_basics.jl
+++ b/test/test_basics.jl
@@ -26,7 +26,7 @@ elts = (Float32, Float64, ComplexF32, ComplexF64)
 @testset "KroneckerArrays (eltype=$elt)" for elt in elts
   p = [1, 2] × [3, 4, 5]
   @test length(p) == 6
-  @test collect(p) == [1 × 3, 2 × 3, 1 × 4, 2 × 4, 1 × 5, 2 × 5]
+  @test collect(p) == [1 × 3, 1 × 4, 1 × 5, 2 × 3, 2 × 4, 2 × 5]
 
   r = @constinferred cartesianrange(2, 3)
   @test r ===
@@ -39,10 +39,10 @@ elts = (Float32, Float64, ComplexF32, ComplexF64)
   @test first(r) == 1
   @test last(r) == 6
   @test r[1 × 1] == 1
-  @test r[2 × 1] == 2
-  @test r[1 × 2] == 3
-  @test r[2 × 2] == 4
-  @test r[1 × 3] == 5
+  @test r[1 × 2] == 2
+  @test r[1 × 3] == 3
+  @test r[2 × 1] == 4
+  @test r[2 × 2] == 5
   @test r[2 × 3] == 6
 
   r = @constinferred(cartesianrange(2 × 3, 2:7))
@@ -53,10 +53,10 @@ elts = (Float32, Float64, ComplexF32, ComplexF64)
   @test first(r) == 2
   @test last(r) == 7
   @test r[1 × 1] == 2
-  @test r[2 × 1] == 3
-  @test r[1 × 2] == 4
-  @test r[2 × 2] == 5
-  @test r[1 × 3] == 6
+  @test r[1 × 2] == 3
+  @test r[1 × 3] == 4
+  @test r[2 × 1] == 5
+  @test r[2 × 2] == 6
   @test r[2 × 3] == 7
 
   # Test high-dimensional materialization.
diff --git a/test/test_blocksparsearrays.jl b/test/test_blocksparsearrays.jl
index bd88d9d..770e216 100644
--- a/test/test_blocksparsearrays.jl
+++ b/test/test_blocksparsearrays.jl
@@ -1,10 +1,10 @@
 using Adapt: adapt
 using BlockArrays: Block, BlockRange, mortar
 using BlockSparseArrays:
-  BlockSparseArray, BlockSparseMatrix, blockrange, blocksparse, blocktype
+  BlockIndexVector, BlockSparseArray, BlockSparseMatrix, blockrange, blocksparse, blocktype
 using FillArrays: Eye, SquareEye
 using JLArrays: JLArray
-using KroneckerArrays: KroneckerArray, ⊗, ×
+using KroneckerArrays: KroneckerArray, ⊗, ×, arg1, arg2
 using LinearAlgebra: norm
 using MatrixAlgebraKit: svd_compact
 using Test: @test, @test_broken, @testset
@@ -48,7 +48,18 @@ arrayts = (Array, JLArray)
   @test a[Block(2, 2)[(2:3) × (2:3), (2:3) × (2:3)]] ==
     a[Block(2, 2)][(2:3) × (2:3), (2:3) × (2:3)]
   @test a[Block(2, 2)[(:) × (2:3), (:) × (2:3)]] == a[Block(2, 2)][(:) × (2:3), (:) × (2:3)]
-  @test_broken a[Block(2, 2)][(1:2) × (2:3), (:) × (2:3)]
+  @test a[Block(2, 2)[(1:2) × (2:3), (:) × (2:3)]] ==
+    a[Block(2, 2)][(1:2) × (2:3), (:) × (2:3)]
+
+  # Blockwise slicing, shows up in truncated block sparse matrix factorizations.
+  I1 = BlockIndexVector(Block(1), Base.Slice(Base.OneTo(2)) × [1])
+  I2 = BlockIndexVector(Block(2), Base.Slice(Base.OneTo(3)) × [1, 3])
+  I = [I1, I2]
+  b = a[I, I]
+  @test b[Block(1, 1)] == a[Block(1, 1)[(1:2) × [1], (1:2) × [1]]]
+  @test iszero(b[Block(2, 1)])
+  @test iszero(b[Block(1, 2)])
+  @test b[Block(2, 2)] == a[Block(2, 2)[(1:3) × [1, 3], (1:3) × [1, 3]]]
 
   # Slicing
   r = blockrange([2 × 2, 3 × 3])
@@ -159,7 +170,22 @@ end
   @test a[Block(2, 2)[(2:3) × (2:3), (2:3) × (2:3)]] ==
     a[Block(2, 2)][(2:3) × (2:3), (2:3) × (2:3)]
   @test a[Block(2, 2)[(:) × (2:3), (:) × (2:3)]] == a[Block(2, 2)][(:) × (2:3), (:) × (2:3)]
-  @test_broken a[Block(2, 2)][(1:2) × (2:3), (:) × (2:3)]
+  @test a[Block(2, 2)[(1:2) × (2:3), (:) × (2:3)]] ==
+    a[Block(2, 2)][(1:2) × (2:3), (:) × (2:3)]
+
+  # Blockwise slicing, shows up in truncated block sparse matrix factorizations.
+  I1 = BlockIndexVector(Block(1), Base.Slice(Base.OneTo(2)) × [1])
+  I2 = BlockIndexVector(Block(2), Base.Slice(Base.OneTo(3)) × [1, 3])
+  I = [I1, I2]
+  b = a[I, I]
+  @test b[Block(1, 1)] == a[Block(1, 1)[(1:2) × [1], (1:2) × [1]]]
+  @test arg1(b[Block(1, 1)]) isa Eye
+  @test iszero(b[Block(2, 1)])
+  @test arg1(b[Block(2, 1)]) isa Eye
+  @test iszero(b[Block(1, 2)])
+  @test arg1(b[Block(1, 2)]) isa Eye
+  @test b[Block(2, 2)] == a[Block(2, 2)[(1:3) × [1, 3], (1:3) × [1, 3]]]
+  @test arg1(b[Block(2, 2)]) isa Eye
 
   # Slicing
   r = blockrange([2 × 2, 3 × 3])