Is it permissible for me to keep the wages for work I do outside the workshop without giving my partner anything, knowing that his contribution is financial, and my contribution is in labor and management, with our agreement to split the profits equally?

If the workshop belongs to your relative and you work in it and the profit is split equally, then this partnership is valid according to the Hanbali school, unlike the majority of scholars. The Hanbalis believe that if a man gives his animal to another to work with it and the profit is shared between them, it is valid. However, the majority of scholars consider it invalid, believing that all profit belongs to the owner of the animal, and the worker is entitled to a wage similar to what he would normally earn. The reasoning of the Hanbalis is that it is an asset that grows through work, so a contract for a share of its growth is valid, similar to dirhams and dinars, and trees in musaqat (sharecropping of trees), resembling musaqat and muzara'ah (sharecropping of land). If the agreement is as described, then the owner of the workshop is only entitled to what was done inside it or with his equipment. As for what you did outside the workshop and not with equipment owned by your relative, its wage is entirely yours, while observing the agreement between you and informing your relative of your work outside the workshop as a precaution.