Thank you very much for your comments. I actually only wanted this to be my avatar and not a "pixel art piece", haha. The white bg is because I want it in my avatar, and the lack of external AA is because this is used in different backgrounds, but I might rework the external line, though I'd like to keep the 4 color restriction.
But this white background is absolutely necessary to the piece!
Agree with Jeremy, since there is a bg.
Great piece!
This would look way better with some external AA in the greys.
It's preferred you add transparency if there is no significance in using a white background.
In that case, I do agree with adding more AA (on the outer contours especially). AA is completely possible with your current palette, so you shouldn't have much trouble doing so.