To reproduce:

Add a RadCommandBar with a row and a strip. Add a CommandBarSplitButton and subscribe to its Click event. You will see that when you click the arrow button the event will be fired. In result you cannot distinguish whether the arrow button is clicked.

Workaround:

private bool arrowButtonClicked;
void ArrowPart_Click(object sender, EventArgs e)
{
    this.arrowButtonClicked = true;

    //... CommandBarSplitButton.ArrowPart logic
}

void button_Click(object sender, EventArgs e)
{
    if (this.arrowButtonClicked)
    {
        this.arrowButtonClicked = false;
        return;
    }

    //...CommandBarSplitButton logic
}