Can a multiclass Wizard/Fighter (Eldritch Knight) add EK known spells to the wizard spellbook?

Question

When intending to play a Fighter (Eldritch Knight) I was dissatisfied with the spell restrictions so I devised a plan to increase my spell versatility:

1. Start with 1 wizard level to have a spellbook with 6 wizard lvl 1 spells
2. Take all the other levels as fighter, picking eldritch knight at fighter 3 (total character lvl 4) and adding all spells learned at this time to the wizard spellbook.
3. Each subsequent fighter level, replace the "any school" spell learned (this has been clarified to be possible, see this related answer) and keep adding it to the wizard spellbook. This effectively allows 1 extra spell to be added to the book per fighter level after 3.

The only dubious part of this plan is the possibility to add EK known spells to the wizard spellbook. So can this be done by RAW?

Answer 1

The intent is likely yes

Firstly, the Eldritch Knight spells are actually wizard spells (Arcane Trickster uses similar wording):

You know three 1st level wizard spells of your choice,

However, it is not explicitly stated that knowing a spell counts as finding it for the purposes of adding it to a spellbook. From the wizard spellbook section:

You might find other spells during your adventures. You could discover a spell recorded on a scroll in an evil wizard’s chest, for example, or in a dusty tome in an ancient library.

Let us then make a comparison with another feature that also allows one to "know" wizard spells (as opposed to having them on the spellbook), the Magic Initiate feat.

In this sage advice we can see under a somewhat unrelated question about the Magic Initiate Feat:

Similarly, if you are a wizard and pick that class for the feat, you learn a 1st-level wizard spell, which you could add to your spellbook and subsequently prepare.

To me this part of the sage advice answer reads as a clarification of the general rules and thus I interpret that a wizard spell known could normally be added to a wizard spellbook (if you are a wizard), also allowing spells known from Eldritch Knight (or Arcane Trickster) to be added in this way.

Naturally, one could also read this as Magic Initiate being an exception to the general rules, in which case this strategy would not be possible.

However, regardless of how you choose to interpret this...

You are unlikely to break anything by allowing this particular build

My DM did allow me to make such a character and having played the character up to total level 10 (wizard 1/EK 9) I can note the following:

• All fighter features were obtained 1 level later than a pure EK
• Only lvl 1 spells could be copied in this way, and I could only prepare INT+1 lvl 1 wizard spells. After a few levels only a few lvl 1 spells remain relevant.
• I could not replace any other EK spell (I technically could but would miss out on getting a new spell)
• I found this mostly useful to have a lot of rituals available but in hindsight that might have been similarly accomplished by simply taking the Ritual Caster feat.

The extra cantrips and rituals were the main features that ended up being relevant, mainly because I was the only caster in the group. There seemed to be no impact on game balance.