The reason you cannot is because what you do to others, you do to yourself.
While you're being angry towards the other person, YOU are suffering from that anger as well. You might be feeling righteous or justified, but you're not feeling love. And then you might say, well, when I'm not angry at this person for a short while, I can go back to loving ME because I'm innocent, but they're not. And then you flip-flop back and forth. And you think that you love yourself but the unhealed grievance STAYS with you and lurks in the background, undermining your own self-love.
The ego's trick is to make you believe you can have an attitude about someone else that does not apply to you. That there is a separation between the two of you, and therefore they can be wrong while you are right, or your anger is about them and not about yourself. Mistake.
You can't love yourself fully if you don't love others, and vice versa. You can't pick and choose. Either you share love for the benefit of everyone or you are attacking everyone, and in both cases that includes you.
So try it next time you're mad at someone or upset by someone... see if you can feel complete self-love at the same time. You won't be able to, because you're being unloving to yourself.
If it were true that we are all separate, then you would have to have the ability to love and hate at the same time. That is clearly impossible to any rational person.
We are all one.