al-Salāmu ‘alaykum wa rahmatullāh,
The verb خالَ khaala has two different forms that also differ in pattern and meaning.
The first is the verb خال khaala (perfect tense) يَخُولُ yakhoolu (imperfect tense), خَولا khawlan (verbal noun), and it means ‘to do proficiently’ or ‘to perfect’. One may use it in the phrase خَوَّلَهُ اللهُ نِعمةً مِنْ عِنْدِهِ khawwalahu Allaahu ni3matan min 3indihi to mean ‘Allaah [proficiently] bestowed upon him blessings from Himself.’
This meaning also allows us to recognise the importance and status of the maternal uncle and aunt, and indeed our obligations as maternal aunts and uncles, who are called the خَالٌ khaal and the خالَةٌ khaalah because they are supposed to ‘take care proficiently’ of their family. And this may be one reason why the maternal aunt in Islam is afforded the status of the mother when the mother is absent.
Allaah used it in this way in the Qur’an,
وَتَرَكْتُم مَّا خَوَّلْنَاكُمْ وَرَاء ظُهُورِكُمْ
and you have left whatever We bestowed upon you behind you (al-An’aam, verse 94 )
The second form is the verb خالَ khaala (perfect tense) يَخالُ yakhaalu (imperfect tense) and it has two separate meanings. The first means ‘he came to possess much wealth’ including slaves, chattel and servants. The second meaning is ‘he supposed’ and it is one of the sisters of the verb طَنَّ dhanna.
Allaah also used the verb according to this meaning in the Qur’an, (Ta-Ha, verse 66),
فإذا حبالهم وعصيهم يخيل إليه من سحرهم أنها تسعى
And suddenly their ropes and staffs seemed to him from their magic that they were moving [like snakes]
and in Surah Luqman, verse 18,
إن الله لا يحب كل مختال فخور
Indeed, Allah does not like everyone self-deluded and boastful.
Here, the word مًَُخْتال mukhtaal (which is the active participle from the verb اخْتالَ ikhtaala which is extremely closely related to the verb خالَ khaala) has been translated as self-deluded and as such is related to the first meaning of coming into possession of wealth, for too much of a good thing leads one to arrogance and feelings that they are self-sufficient.
Allaah also used a derivative of this word in (al-Nahl, verse 8 )
والخيل والبغال والحمير لتركبوها وزينة ويخلق ما لا تعلمون
And [He created] the horses, mules and donkeys for you to ride and [as] adornment.
So what does a horse have to do with any of the meanings conveyed above? The same question was asked to Abu Hatim, one of the classical scholars of Arabic, but he was unable to answer. It was reported that a madman was walking by and heard the question, so he said, “I will tell you! The horse was called a خيل khayl due to the pride and arrogance (اختيال ikhtiyaal - the verbal noun from the verb اختال ikhtaala) it displays when it walks!” Upon which Abu Hatim said, “Note down this wise reply and pass on the knowledge, even if it is on the authority of a madman!”
A search in the dictionary al-Qamus al-Muhit shows that even al-Fayroozabadi wrote that the word خَيْل khayl means pride, arrogance, and vanity, so the madman’s reply could very well be how the horse got his name.