Is the valediction "see you later, alligator" used in English?

I have been reading a book about real English expressions and idioms. The first chapter deals with greetings. According to the section On departure,

See you later, alligator

is one of the informal ways of saying goodbye to someone.

The book says that it's humoristic and the answer would be in a while crocodile. Apparently, it's derived from a 50s pop song.

My question is: Is this expression common? It goes without saying that I am not a native speaker of English but I usually listen to English music, films and TV shows and I have never heard of it before.

The line is from a 50s rock n roll song. You can watch Bill Haley and the Comets performing this little ditty on Youtube See you later, Aligator

It was a catchy line and it caught on in popularity, and I'd say it resisted until the late 60s until it gradually declined in usage. Ironically, this form of greeting is seen as being quaint and/or painfully "uncool". I doubt if any of today's teenagers would be caught dead saying this.

As far as I know, the expression comes from a song by Bill Haley and the Comets, one of the earliest rock and roll groups, in which the chorus included the words:

See you later alligator

After 'while crocodile

To use it now, as I’m sure some do, seems very dated, unless, of course, it is done in a spirit of irony.

When I was a kid, my grandfather used to say this to my brother and I when we left his house. He would say, "See you later, alligator" and we would say "After awhile, crocodile!" and then he would come back with "By the light of the moon, racoon!", a phrase he just added on.

We were little kids, so that always got us. We could never come up with "Gotta go, buffalo!" or something else to continue the chain.

I don't use the phrase regularly, but in my rather standard American midwest office, I would wager a large sum of money that if I said, "See you later, alligator!" in context to the 30, 40, 50, and 60 somethings that work here, to the person I would get the crocodile response.

I've been using this phrase a lot recently. My Mother, aged 92 with advanced Alzheimer's. When there was almost nothing else left we'd say "See you later!" she'd say "Alligator! In a while... a Crocodile!" Aside from that I've not heard it used in 50 years.

She died this morning. The last thing I said to her was "see you later alligator" and gave her a kiss.