diff --git a/xarray/core/indexing.py b/xarray/core/indexing.py
index f48c9e72af1..d587ee3a02b 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
@@ -1291,6 +1291,24 @@ 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 1c3ab2ec229..8e62341d5ee 100644
--- a/xarray/core/variable.py
+++ b/xarray/core/variable.py
@@ -5,7 +5,7 @@
 from collections import defaultdict
 from datetime import timedelta
 from distutils.version import LooseVersion
-from typing import Any, Dict, Hashable, Iterable, Mapping, TypeVar, Union
+from typing import Any, Dict, Hashable, Mapping, TypeVar, Union
 
 import numpy as np
 import pandas as pd
@@ -540,20 +540,6 @@ def _broadcast_indexes(self, key):
             k.item() if isinstance(k, np.ndarray) and k.ndim == 0 else k for k in key
         )
 
-        key_dict = dict(zip(self.dims, key))
-        for dim, k in key_dict.items():
-            if isinstance(k, Iterable):
-                # let da.sel(x=da.x) pass but skip if Variable has different dimensions
-                # e.g. da.sel(x=Variable(("points",), [0, 1, 2]))
-                if isinstance(k, Variable) and k.dims != (dim,):
-                    continue
-                if duck_array_ops.array_equiv(k, np.arange(self.sizes[dim])):
-                    # 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)
-                    key_dict[dim] = slice(None)
-        key = tuple(key_dict.values())
-
         if all(isinstance(k, BASIC_INDEXING_TYPES) for k in key):
             return self._broadcast_indexes_basic(key)