diff --git a/Project.toml b/Project.toml
index ea6297b8..2354043c 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "ManifoldsBase"
 uuid = "3362f125-f0bb-47a3-aa74-596ffd7ef2fb"
 authors = ["Seth Axen <seth.axen@gmail.com>", "Mateusz Baran <mateuszbaran89@gmail.com>", "Ronny Bergmann <manopt@ronnybergmann.net>", "Antoine Levitt <antoine.levitt@gmail.com>"]
-version = "0.13.12"
+version = "0.13.13"
 
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
diff --git a/src/ManifoldsBase.jl b/src/ManifoldsBase.jl
index f1534032..1ac56dc9 100644
--- a/src/ManifoldsBase.jl
+++ b/src/ManifoldsBase.jl
@@ -77,6 +77,17 @@ function allocate(::AbstractManifold, a, T::Type, dim1::Integer, dims::Integer..
 end
 allocate(::AbstractManifold, a, T::Type, dims::Tuple) = allocate(a, T, dims)
 
+"""
+    _pick_basic_allocation_argument(::AbstractManifold, f, x...)
+
+Pick which one of elements of `x` should be used as a basis for allocation in the
+`allocate_result(M::AbstractManifold, f, x...)` method. This can be specialized to, for
+example, skip `Identity` arguments in Manifolds.jl group-related functions.
+"""
+function _pick_basic_allocation_argument(::AbstractManifold, f, x...)
+    return x[1]
+end
+
 """
     allocate_result(M::AbstractManifold, f, x...)
 
@@ -88,7 +99,8 @@ isomorphisms.
 """
 @inline function allocate_result(M::AbstractManifold, f, x...)
     T = allocate_result_type(M, f, x)
-    return allocate(M, x[1], T)
+    picked = _pick_basic_allocation_argument(M, f, x...)
+    return allocate(M, picked, T)
 end
 @inline function allocate_result(M::AbstractManifold, f)
     T = allocate_result_type(M, f, ())
diff --git a/src/PowerManifold.jl b/src/PowerManifold.jl
index 5a1cd573..478ca72a 100644
--- a/src/PowerManifold.jl
+++ b/src/PowerManifold.jl
@@ -130,8 +130,11 @@ const PowerManifoldNestedReplacing = AbstractPowerManifold{
     NestedReplacingPowerRepresentation,
 } where {𝔽}
 
-_access_nested(x, i::Int) = x[i]
-_access_nested(x, i::Tuple) = x[i...]
+# _access_nested(::AbstractManifold, x, i::Tuple) can be overloaded to achieve
+# manifold-specific nested element access (for example to `Identity` on power manifolds).
+@inline _access_nested(M::AbstractManifold, x, i::Int) = _access_nested(M, x, (i,))
+@inline _access_nested(::AbstractManifold, x, i::Tuple) = _access_nested(x, i)
+@inline _access_nested(x, i::Tuple) = x[i...]
 
 function Base.:^(
     M::PowerManifold{
@@ -150,7 +153,7 @@ function allocate_result(M::PowerManifoldNested, f, x...)
         return allocate(M, x[1])
     else
         return [
-            allocate_result(M.manifold, f, map(y -> _access_nested(y, i), x)...) for
+            allocate_result(M.manifold, f, map(y -> _access_nested(M, y, i), x)...) for
             i in get_iterator(M)
         ]
     end
@@ -199,7 +202,9 @@ function allocate_result(
     )
 end
 function allocate_result(M::PowerManifoldNested, f::typeof(get_vector), p, X)
-    return [allocate_result(M.manifold, f, _access_nested(p, i)) for i in get_iterator(M)]
+    return [
+        allocate_result(M.manifold, f, _access_nested(M, p, i)) for i in get_iterator(M)
+    ]
 end
 function allocate_result(::PowerManifoldNestedReplacing, ::typeof(get_vector), p, X)
     return copy(p)
@@ -452,7 +457,7 @@ function get_coordinates(
             M.manifold,
             _read(M, rep_size, p, i),
             _read(M, rep_size, X, i),
-            _access_nested(B.data.bases, i),
+            _access_nested(M, B.data.bases, i),
         ) for i in get_iterator(M)
     ]
     return reduce(vcat, reshape(vs, length(vs)))
@@ -492,7 +497,7 @@ function get_coordinates!(
             view(c, v_iter:(v_iter + dim - 1)),
             _read(M, rep_size, p, i),
             _read(M, rep_size, X, i),
-            _access_nested(B.data.bases, i),
+            _access_nested(M, B.data.bases, i),
         )
         v_iter += dim
     end
@@ -532,7 +537,7 @@ function get_vector!(
             _write(M, rep_size, Y, i),
             _read(M, rep_size, p, i),
             c[v_iter:(v_iter + dim - 1)],
-            _access_nested(B.data.bases, i),
+            _access_nested(M, B.data.bases, i),
         )
         v_iter += dim
     end
@@ -553,7 +558,7 @@ function get_vector!(
             M.manifold,
             _read(M, rep_size, p, i),
             c[v_iter:(v_iter + dim - 1)],
-            _access_nested(B.data.bases, i),
+            _access_nested(M, B.data.bases, i),
         )
         v_iter += dim
     end
@@ -604,7 +609,7 @@ function _get_vectors(
     rep_size = representation_size(M.manifold)
     vs = typeof(zero_tv)[]
     for i in get_iterator(M)
-        b_i = _access_nested(B.data.bases, i)
+        b_i = _access_nested(M, B.data.bases, i)
         p_i = _read(M, rep_size, p, i)
         for v in b_i.data
             new_v = copy(M, p, zero_tv)
@@ -622,7 +627,7 @@ function _get_vectors(
     zero_tv = zero_vector(M, p)
     vs = typeof(zero_tv)[]
     for i in get_iterator(M)
-        b_i = _access_nested(B.data.bases, i)
+        b_i = _access_nested(M, B.data.bases, i)
         for v in b_i.data
             new_v = copy(M, p, zero_tv)
             new_v[i...] = v