By saying his brother, John is referring to two Christians.
False Christian is not mentioned.
But it is implied. John is
assuming that the person is a Christian. As we are to do to somebody who tells us he is a Christian. It is possible for a person to believe in vain.
1 Corinthians 15:1 Moreover, brethren, I declare to you the gospel which I preached to you, which also you received and in which you stand,
2 by which also you are saved, if you hold fast that word which I preached to you—
unless you believed in vain.
Vain means
Not yielding the desired outcome; fruitless.
Lacking substance or worth
Act 8:13 Then Simon himself also believed; and when he was baptized he continued with Philip, and was amazed, seeing the miracles and signs which were done
then later Peter says to him
Act 8:21 You have neither part nor portion in this matter, for
your heart is not right in the sight of God.
So, just claiming to believe is not enough.
1 John_2:4 He who says, "I know Him," and does not keep His commandments,
is a liar, and the truth is not in him.
1 John_4:20 If someone says, "I love God," and hates his brother, he
is a liar
The Christian who hates his brother started off being a brother in Christ, then he no longer has eternal life abiding in him.
Again, you are making a category mistake.
You are mistaking John calling somebody a brother with that person being born again.
John calls him a brother out of charity. That person claims to be a brother, so John calls him that.
We can take the words of 1 John 4:20 and just change a few words and get the same meaning.
1 John_4:20 If someone says, "I love God," and hates his brother,
he is a liar
If someone claims to be a brother and hates his brother,
he is a liar
If someone claims to be a Christian and hates his brother,
he is a liar
You don't become born again and then at some future time start to hate your brother and become un-born again.