diff --git a/xarray/core/indexing.py b/xarray/core/indexing.py
index 4e58be1ad2f..ab049a0a4b4 100644
--- a/xarray/core/indexing.py
+++ b/xarray/core/indexing.py
@@ -4,7 +4,7 @@
 from collections import defaultdict
 from contextlib import suppress
 from datetime import timedelta
-from typing import Any, Callable, Sequence, Tuple, Union
+from typing import Any, Callable, Iterable, Sequence, Tuple, Union
 
 import numpy as np
 import pandas as pd
@@ -1314,6 +1314,24 @@ def __init__(self, array):
         self.array = array
 
     def __getitem__(self, key):
+
+        if not isinstance(key, VectorizedIndexer):
+            # if possible, short-circuit when keys are effectively slice(None)
+            # This preserves dask name and passes lazy array equivalence checks
+            # (see duck_array_ops.lazy_array_equiv)
+            rewritten_indexer = False
+            new_indexer = []
+            for idim, k in enumerate(key.tuple):
+                if isinstance(k, Iterable) and duck_array_ops.array_equiv(
+                    k, np.arange(self.array.shape[idim])
+                ):
+                    new_indexer.append(slice(None))
+                    rewritten_indexer = True
+                else:
+                    new_indexer.append(k)
+            if rewritten_indexer:
+                key = type(key)(tuple(new_indexer))
+
         if isinstance(key, BasicIndexer):
             return self.array[key.tuple]
         elif isinstance(key, VectorizedIndexer):
diff --git a/xarray/core/variable.py b/xarray/core/variable.py
index 058b7bf52d4..daa8678157b 100644
--- a/xarray/core/variable.py
+++ b/xarray/core/variable.py
@@ -1057,7 +1057,9 @@ def isel(
 
         invalid = indexers.keys() - set(self.dims)
         if invalid:
-            raise ValueError("dimensions %r do not exist" % invalid)
+            raise ValueError(
+                f"dimensions {invalid} do not exist. Expected one or more of {self.dims}"
+            )
 
         key = tuple(indexers.get(dim, slice(None)) for dim in self.dims)
         return self[key]
diff --git a/xarray/tests/test_dask.py b/xarray/tests/test_dask.py
index cc554850839..8fb54c4ee84 100644
--- a/xarray/tests/test_dask.py
+++ b/xarray/tests/test_dask.py
@@ -1390,3 +1390,58 @@ def test_lazy_array_equiv_merge(compat):
             xr.merge([da1, da3], compat=compat)
     with raise_if_dask_computes(max_computes=2):
         xr.merge([da1, da2 / 2], compat=compat)
+
+
+@pytest.mark.filterwarnings("ignore::FutureWarning")  # transpose_coords
+@pytest.mark.parametrize("obj", [make_da(), make_ds()])
+@pytest.mark.parametrize(
+    "transform",
+    [
+        lambda a: a.assign_attrs(new_attr="anew"),
+        lambda a: a.assign_coords(cxy=a.cxy),
+        lambda a: a.copy(),
+        lambda a: a.isel(x=np.arange(a.sizes["x"])),
+        lambda a: a.isel(x=slice(None)),
+        lambda a: a.loc[dict(x=slice(None))],
+        lambda a: a.loc[dict(x=np.arange(a.sizes["x"]))],
+        lambda a: a.loc[dict(x=a.x)],
+        lambda a: a.sel(x=a.x),
+        lambda a: a.sel(x=a.x.values),
+        lambda a: a.transpose(...),
+        lambda a: a.squeeze(),  # no dimensions to squeeze
+        lambda a: a.sortby("x"),  # "x" is already sorted
+        lambda a: a.reindex(x=a.x),
+        lambda a: a.reindex_like(a),
+        lambda a: a.rename({"cxy": "cnew"}).rename({"cnew": "cxy"}),
+        lambda a: a.pipe(lambda x: x),
+        lambda a: xr.align(a, xr.zeros_like(a))[0],
+        # assign
+        # swap_dims
+        # set_index / reset_index
+    ],
+)
+def test_transforms_pass_lazy_array_equiv(obj, transform):
+    with raise_if_dask_computes():
+        assert_equal(obj, transform(obj))
+
+
+def test_more_transforms_pass_lazy_array_equiv(map_da, map_ds):
+    with raise_if_dask_computes():
+        assert_equal(map_ds.cxy.broadcast_like(map_ds.cxy), map_ds.cxy)
+        assert_equal(xr.broadcast(map_ds.cxy, map_ds.cxy)[0], map_ds.cxy)
+        assert_equal(map_ds.map(lambda x: x), map_ds)
+        assert_equal(map_ds.set_coords("a").reset_coords("a"), map_ds)
+        assert_equal(map_ds.update({"a": map_ds.a}), map_ds)
+
+        # fails because of index error
+        # assert_equal(
+        #     map_ds.rename_dims({"x": "xnew"}).rename_dims({"xnew": "x"}), map_ds
+        # )
+
+        assert_equal(
+            map_ds.rename_vars({"cxy": "cnew"}).rename_vars({"cnew": "cxy"}), map_ds
+        )
+
+        assert_equal(map_da._from_temp_dataset(map_da._to_temp_dataset()), map_da)
+        assert_equal(map_da.astype(map_da.dtype), map_da)
+        assert_equal(map_da.transpose("y", "x", transpose_coords=False).cxy, map_da.cxy)