Learn Arabic (Modern Standard): قطع أخي إصبعه بسكين المطبخ اليوم. - My brother cut his finger with the kitchen knife today.

The pronoun suffix ـه means 'his' and refers back to the possessor (أخي, my brother). In Arabic, possessive suffixes always match the gender of the owner, not the object being owned. (Also, while paired body parts like hands are feminine, إصبع can actually be treated as masculine or feminine, but the suffix ـه here is strictly about the brother.)

How do we know the grammatical case of أخي (my brother) since there are no visible case endings?

أخي is the subject of the verb قطع, so it is in the nominative case (marfūʿ). However, when any noun is attached to the 'my' suffix (ـي), its final vowel is forced to be a kasra to blend smoothly with the ي sound. This hides the usual damma, giving it an 'implied' case ending.

Why doesn't سكين (knife) have the definite article الـ if the translation says 'the kitchen knife'?

سكين المطبخ is an iḍāfa (possessive construction) meaning 'the knife of the kitchen'. In an iḍāfa, the first word (سكين) never takes الـ. It becomes definite automatically because the second word (المطبخ) has the definite article.

Why does اليوم (today) take a fatḥa at the end in this sentence?

Here, اليوم is acting as an adverb of time, telling us when the action happened. In Arabic, adverbs of time and place take the accusative case (manṣūb), which is marked by a fatḥa at the end.