
königlich
royal
Relating to or characteristic of a king, queen, or monarchy.
Die königliche Familie besuchte die Eröffnungszeremonie.
The royal family attended the opening ceremony.
royal
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royal
Relating to or characteristic of a king, queen, or monarchy.
Die königliche Familie besuchte die Eröffnungszeremonie.
The royal family attended the opening ceremony.

royal
Used figuratively for something splendid, luxurious, or first-class. Often used in the phrase 'sich königlich amüsieren' (to have a royal time / to enjoy oneself immensely).
Wir haben uns bei dem Festessen königlich amüsiert.
We enjoyed ourselves immensely at the banquet.
The adjective 'königlich' (royal) is derived directly from the noun 'der König' (the king). The ending '-lich' is often used to form an adjective from a noun.
Das königliche Wappen ziert den Eingang des Palastes.
The royal coat of arms adorns the entrance to the palace.
In its literal meaning, 'königlich' (royal) describes nouns that are directly connected with a monarchy. Common combinations are 'die königliche Familie' (the royal family), 'das königliche Schloss' (the royal palace/castle), or 'königliche Hoheit' (royal highness).
Die Kronjuwelen werden im königlichen Palast aufbewahrt.
The crown jewels are kept in the royal palace.
In the figurative meaning, the phrase 'sich königlich amüsieren' is very common and means that you are having a lot of fun. The word can also intensify other verbs of enjoyment, such as 'königlich speisen' (to dine like royalty).
Die Kinder haben sich auf dem Geburtstagfest königlich amüsiert.
The children had a wonderful time at the birthday party.
princely / lavish
Both words can figuratively emphasize something especially generous or luxurious. königlich is particularly appropriate for a splendid, first-class experience, while fürstlich often describes exceptionally generous pay or amenities. So königlich is more suitable for a luxurious experience, while fürstlich is more suitable for a generous reward.

In diesem erstklassigen Hotel wurden wir königlich verwöhnt.
We were treated like royalty at this first-class hotel.
Für seine erfolgreiche Arbeit wurde er fürstlich entlohnt.
He was handsomely rewarded for his successful work.