Split pairs
Wow! Where to start?
First, splitting pairs and doubling down are two different things.
Splitting pairs is when your first two cards are the same rank, you can split them by putting up a seperate bet, get an additional card on each, and play each newly formed hand seperately (and one of the things you can do sometimes, is double down on one of these new hands). If you again receive a same ranked card on one of these newly formed hands, most places allow you to split again, until you have reached a limit of four seperate hands.
Doubling Down is when you match your original bet with an additional wager, and by being able to bet more after you have seen your cards and the dealer's up card, you receive one additional card on your hand. You cannot play your hand any further after the double down and must stand at that point. You are playing one hand, getting one card, for the oppertunity of increasing your bet. Most places also allow you to double down for less money than your original bet, although this is a bad option. Doubling Down for more is generally frowned upon by the casino.
I suggest you re-read BlackJack Secrets, or maybe you didn't state your question clearly and I don't understand it.
When do you split pairs? Depends on what up-card the dealer has.
In general:
Split 2s when the dealer is showing 2 thru 7
Split 3s when the dealer is showing 2 thru 7
Only split 4s when the dealer is showing 5 or 6
Never split 5s (that is 10, you probably should be doubling down).
Split 6s when the dealer is showing 2 thru 6
Split 7s when the dealer is showing 2 thru 7
Always split 8s.
Split 9s when the dealer is showing 2 thru 9 (except when he has 7 showing).
Never Split 10s. (20 is a great hand, why mess it up?)
Always split Aces.
That is the general rules. The exact details on one or two can change depending on the number of decks they are using, if the dealer hits or stands on soft 17, or if you are allowed to double after splitting.
As an example, you would not split 4s, even against a dealer's 5 or 6, if double after split is not allowed. You would just play the hand as an 8 and hit it.