A sermon on the Transfiguration by a Frankish abbot of a monastery in southern Italy named Autpert Ambrose (d. 784) discusses why Jesus took only Peter, James, and John with him when he was to be transfigured:

"Why did he [the Lord] go up on the mountain to pray with only three disciples? Why did he manifest the glory of his majesty to these three only at that time?  Shall it be thought that he did this without any reason, that he hid the mystery of his revelation from the other Apostles, to whom the Holy Scriptures attest he said: 'To you it is given to know the mystery of God' (Luke 8, 10)?  Or, again: 'All that I have heard from the Father I have made known to you' (John 15, 15)?  Let us grant, then, that these were the primates, the "pillars" of the Apostles, as Paul spoke of Peter, James, and John, saying that they 'seemed to be the pillars' (Galatians 2, 9).  Therefore, The Lord revealed his secrets rather to these than to the others: to these alone apart he revealed the sadness of his heart concerning the betrayal of Judas and the Jews, which was soon to take place in his Passion.  Shall someone think that this mystery was a secret?  By no means.  Peter, James, and John alone were taken up for the teaching of the great glory of the Redeemer through its prefiguring.  Now, the name 'Peter' is taken from the word 'petra' [rock].  But Paul taught that, 'Christ was the rock' (1 Corinthians 10, 4).  Peter said to this Rock: 'You are Christ, the Son of the living God' (Matthew 16, 16).  The faith of the whole Church was spoken truly in these words, and so how fittingly the Rock said to the rock: 'I say to you that you are Peter, and upon this rock I will build my Church.'  That is, The Lord said clearly to him, I will build you upon me -- you, who are the figure of the Church, which will be built on you and upon me.  Thus, when the same blessed Peter, giving prophetic testimony, said: 'Behold, I set in Sion a cornerstone that is precious and elect as a foundation', he joined to this the words: 'Be you as living stones, built up into spiritual houses' (1 Peter 2, 5-6).  Likewise also, when Paul said: 'No one can lay a foundation other than the one that has been laid, which is Christ Jesus', he immediately added: 'Let each man see how he builds upon it' (1 Corinthians 3, 11).  Therefore, it is said that when Peter is set as the foundation, the whole Church is placed upon Christ.  Now, the name 'James' means 'supplanter' in our language.  But what does he supplant?  Vices.  Certainly, we supplant vices in the person of James when we free ourselves with the firstborn of the virtues through the holy zeal of our prayers.  The Psalmist speaks of this supplant action of the vices, saying: 'Blessed is he who shall hold and dash his little ones against the rock' (Psalm 136, 9).  The name 'John' is interpreted 'dove', through whom, without doubt, those are prefigured who are inflamed by the love of God and neighbor within the Holy Church of God.  The 'dove' signifies the Holy Spirit, who is shown and written to be the love of the Father and the Son: 'The love of God is spread forth into our hearts through the Holy Spirit who is given to us' (Romans 5, 5).  Therefore, faith is shown in Peter, the performing of virtues in James, and love in John the Apostle.  It is not unfitting for only these three to ascend the mountain with The Lord, for they bear the figures of all the saints in their own persons."