Lion's Eye Diamond can only generate mana for you in between casting spells, activating abilities, etc.; you can't do it while you're casting a spell
Technically you *can* generate mana while you are casting a spell, however, you cannot pay costs before you have activated all mana abilities. Consider rules 601.2a-g which describes the order of what must be done to play a spell. In particular, look at step a, f and g below:
601.2a. The player announces that he or she is casting the spell. That card (or that copy of a card) moves from the zone it's in to the stack. It becomes the topmost object on the stack. It has all the characteristics of the card (or the copy of a card) associated with it, and that player becomes its controller. The spell remains on the stack until it's countered, it resolves, or an effect moves it elsewhere.
601.2f. If the total cost includes a mana payment, the player then has a chance to activate mana abilities (see rule 605, "Mana Abilities").Mana abilities must be activated before costs are paid.
601.2g. The player pays the total cost in any order. Partial payments are not allowed. Unpayable costs can't be paid.
Once you have announced the spell ala
601.2a, you are in the midst of casting that spell. It's not until step 'f' that you actually need to generate the mana. Pay particular attention to the part of rule
601.2f that I highlighted in bold. Since Lion's Eye Diamond is a mana ability, you must activate it first before you can move on to step 'g' and pay the total cost of a retraced Flame Jab. As mentioned, since part of Lion's Eye Diamond's activation cost is to discard your hand, you will not have a hand left when it comes time to satisfy
601.2g.
Hopefully I didn't come off as an ass in this post. Many people are so accustomed to tapping mana before they announce the spell that they don't know they are actually allowed to do that much later during the process of casting the spell.