The reason for this requirement is that by default the Background property has value of Null. Elements with Null background cannot receive the mouse click events and therefore the SplitButton can't detect that it has to close its DropDown.
If, on the other hand, you want to close the ListBox when you select one of its items you can use the RadSplitButton IsOpen property, which controls the visual state of the DropDownContent. In your case, you can set RadSplitButton.IsOpen to False in the SelectionChanged event handler of the ListBox to close the drop down content when a ListBoxItem is clicked.
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