Ple-Sen
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A clear hierarchy. (We need to decide how the order is managed: either through a round table with the leader primarily serving as an ideological inspiration, a symbol of the organization, and the one who receives more glory and wealth, or the leader is like a high-ranking military officer who commands everyone else. In the first case, we have a knightly brotherhood, and in the second, a semi-military organization).
Any ostentatious element. We should have a motto and a name. Something not too loud (without any dragons), but with ambition. Mentat had something similar.
3. We serve our leader personally; he raised us from rags to knights, and we will rise even higher if he receives a title/land/or something else. This fits perfectly with our leader, and the idea of knights serving a feudal lord is generally suitable. The question is only what the tasks will be and how it will all be presented.
Added 2 minutes ago
All these goals need to be declared in such a way that they can be voiced to anyone who asks why we are getting involved in skirmishes, adventures, and sea voyages. Can you suggest any short and acceptable formulations?
1) By the fact that we have sworn allegiance to a particular country. Therefore, it is our duty as a vassal.
2) By the fact that we are not vassals, but allies of one of the parties in the conflict. Military assistance is part of the agreement.
Added 1 minute ago
Participation in voyages can be justified by an interest in exploring new lands, especially in the third option. Although, I'm not sure if the leader should be sent there.