Where does "сложа" come from in the expression "сложа руки"?

Question

Мы не можем просто сидеть сложа руки.

Given the meaning of the expression -- "sit around, twiddling thumbs", or more literally, "sit around, folding arms" or "sit around with folded arms" -- "сложа" seems to function like an adverbial participle, whether it be present or past, as well as perfective or imperfective:

Мы не можем просто сидеть, сложив руки.

Мы не можем просто сидеть, сложивши руки.

Мы не можем просто сидеть, складывая руки.

... even though the third option doesn't sound right to me. I'm wondering if there are other expressions with a peculiar verb form like "сложа"?

Answer 1

In the Old-Russian language there were 4 past tenses.

All the participles ascend to verbs, past participles ascent to verbs in the past.

All the деепричастия ascend to participles.

So, the perfective деепричастия ascend to the verbs in the past via participles, and some of them inherit for verbs in one past tense, and others for them in the other.

The past tenses in Old-Russian were aorist, imperfect, perfect and plusquamperfect. If you wish, you can find this information, but I believe, that is not what language learners need unless they are studying the history of Russian grammar.

We have retained different деепричастия in our language, and you have just to learn their usage in set or patterned expressions.

Складывая руки means simultaneous actions. It is a present-tense деепричастие. Складывая руки на груди, она говорила.

Сложа руки is a set expression for being idle. Не могу сидеть сложа руки.

Сложив руки means a preceding action. Сложив руки крестообразно, он шагнул вперёд.

Сложивши руки sounds outdated and colloquial.

Answer 2

In fact all alternative options are off since they deviate from the established phraseology.

Can't think of any other but there're other adverbial idioms which use this participle pattern and are still current, namely сломя голову, спустя рукава, положа руку на сердце, скрепя сердце.

I would't call it peculiar, only outdated.

Still more or less acceptable but not necessarily in active use are (without limitation and with varying prefixes)

вспомня, встретя, заметя, наскуча, обнаружа, обратя, оставя, простя, разлюбя, расставя, увидя, услыша, направя, устроя