itashimashita vs. itashimashite

Hello,

I'm trying to figure out which phrase would be correct.

Ryokai shimashita 了解しました
or
Ryokai shimashite 了解しまして

And also...

Ryokai itashimashita 了解いたしました
or
Ryokai itashimashite 了解いたしまして

I keep seeing both used, and I don't know which one is the right one. Also can you explain why one would be right and why the other is wrong? Thank you for any help!

The て form is the usual て form, although making it from the ます form is usually reserved for "super polite" language.

Where did you see/hear it used? The ones that ends with "-ta" is a sentence by itself, but the ones "-te" is just a phrase (something would have to come after that). It would help if you could give the context or the whole sentence or sentences around it.

Off-topic, 了解しました（て） / 了解いたしました（て） is not a really proper way to respond. Correctly, it should be 承知しました（て）/承知いたしました（て）.

了解- is a kind of a colloquial way of responding, and is commonly used among business partners you casually work with, but keep in mind that you should use 承知 to the important people and to clients you just met.