Your request makes sense in this case, but there are scenarios in which the desired behavior will cause side effects, for example if you set a width style via the button's attributes, it will be applied to the arrow element as well, which is not needed.
".k-split-button > .k-primary + .k-split-button-arrow"
On the other hand, CSS styles that need to be targeted at the whole split button (i.e. the split button's wrapper), can be applied via the buttons ID, e.g. #user-toolbar-view.
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