Get a virtual variable (e.g., 'time.year' or a MultiIndex level)
    from a dict of xarray.Variable objects (if possible)

def _get_virtual_variable(variables, key, level_vars=None, dim_sizes=None):
    """Get a virtual variable (e.g., 'time.year' or a MultiIndex level)
    from a dict of xarray.Variable objects (if possible)
    """
    if level_vars is None:
        level_vars = {}
    if dim_sizes is None:
        dim_sizes = {}

    if key in dim_sizes:
        data = pd.Index(range(dim_sizes[key]), name=key)
        variable = IndexVariable((key,), data)
        return key, key, variable

    if not isinstance(key, str):
        raise KeyError(key)

    split_key = key.split('.', 1)
    if len(split_key) == 2:
        ref_name, var_name = split_key
    elif len(split_key) == 1:
        ref_name, var_name = key, None
    else:
        raise KeyError(key)

    if ref_name in level_vars:
        dim_var = variables[level_vars[ref_name]]
        ref_var = dim_var.to_index_variable().get_level_variable(ref_name)
    else:
        ref_var = variables[ref_name]

    if var_name is None:
        virtual_var = ref_var
        var_name = key
    else:
        if _contains_datetime_like_objects(ref_var):
            ref_var = xr.DataArray(ref_var)
            data = getattr(ref_var.dt, var_name).data
        else:
            data = getattr(ref_var, var_name).data
        virtual_var = Variable(ref_var.dims, data)

    return ref_name, var_name, virtual_var