This is one of those examples of the word "believe" being synonymous with "obey"
I agree on this point.
Believe is synonymous with obey, as unbelief is also rendered disobedience. See Hebrews 4:6
Compare KJV with NKJV
Verse six says what the angels' sin was. They left the place God wanted them to be. Jude describes a punishment for them. Then it goes on to the next example of rebellion (or unbelief) by using the phrase, "even as Sodom and Gomorrah". .
And the angels who did not keep their proper domain, but left their own abode, He has reserved in everlasting chains under darkness for the judgment of the great day; as Sodom and Gomorrah, and the cities around them in a similar manner to these, having given themselves over to sexual immorality
The sin of the angels was sexual immorality, as Jude connects the action of the angels with Sodom and Gomorrah.
It's clear from the actual account what happened.
1 Now it came to pass, when men began to multiply on the face of the earth, and daughters were born to them,
2 that the sons of God saw the daughters of men, that they were beautiful; and they took wives for themselves of all whom they chose.
3 And the Lord said, "My Spirit shall not strive with man forever, for he is indeed flesh; yet his days shall be one hundred and twenty years."
4 There were giants on the earth in those days, and also afterward, when the sons of God came in to the daughters of men and they bore children to them. Those were the mighty men who were of old, men of renown. Genesis 6:1-4
These angels left there proper domain and had sexual relations with the women, with the result of producing giants.
Angels leaving there domain and bringing messages or interacting with us, as directed by God is not a sin.
Leaving their proper domain and having sexual relations with women is the sin of sexual immorality that Jude refers to.
Peter also says -
4 For if God did not spare the angels who sinned, but cast them down to hell and delivered them into chains of darkness, to be reserved for judgment;
5 and did not spare the ancient world, but saved Noah, one of eight people, a preacher of righteousness, bringing in the flood on the world of the ungodly;
6 and turning the cities of Sodom and Gomorrah into ashes, condemned them to destruction, making them an example to those who afterward would live ungodly; 2 Peter 2:4-6
Peter teaches us about the ungodly behavior of sexual immorality, and the consequences for such behavior.
Condemned to destruction.
Now today we see such behavior in the world, whereby the interbreeding of humans with other species is become more accepted.
Full Link - http://www.tanahoy.com/psychic-learning/aliens-among-us/
In this day and age, almost everyone has heard or read about aliens – visitors from outer space. Sightings of UFOs are now very common, and because of the Internet, more and more people are able to share their personal experiences with extraterrestrials (ETs). It's undeniable that aliens are real, and these aliens among us are interbreeding with humans.
Alien Child Crossbreed
In fact, they've been coming here for thousands of years, and it’s highly likely that many humans now have alien genes.
So, when we see the phrase from Matthew 24 that says -
But as the days of Noah were, so also will the coming of the Son of Man be. Matthew 24:37
We can see the significance of the sexual immorality being implied by Jude and Peter.
JLB