Not without a jacket, it isn't. How would you feel if men were exposing their underarms "in a business environment". Yours are not any more attractive (in a business environment).
And while the First Lady MAY do that, as she isn't a paid official of the government, I have not seen her do it when she is officially representing her husband, or the nation. The First Lady and the President certainly have the freedom to be as casual as they like, even in swim suits when appropriate. The point of this question seems to hinge on "business environment".
Many things help to reinforce the "glass ceiling". Inappropriate dress by females is one of the things, that will not be mentioned publicly, but is certainly noticed.