Appendix:Proto-Indo-European/ātr-

No. But it would still be good to mention that the laryngeals aren't entirely specified, but that one of them is either h2 or h3.

As mentioned by Benjamin W. Fortson the cover symbol H is used for a laryngeal that cannot or need not be specified, so we can surely use *HeHt(e)r- or *HeHt- for the entry. However, we've 2 other options which can work:

1. *h₂eh₁ter- ("fireplace") - (https://www.academia.edu/16690811/2000_PIE_roots_summary_The_Source_Code_2.4_-_Excel)
2. *hₓehₓtr- ("fire", in *hₓehₓt- at least one laryngeal was h₂ (> *h₂ehₓtr-), perhaps both - p.202) - (https://books.google.com/books?id=tzU3RIV2BWIC) | Michael A. Cahill also lists *hₓehₓtr-.

However Schrijver argues that:

1. "Since it is unlikely that all these forms contain an unmotivated lengthened grade root *h₂et-, a reconstruction *HeHt- is attractive." - (https://books.google.com/books?id=0lNfAAAAMAAJ&focus=searchwithinvolume&q=*HeHt%28e%29r-)
2. ătrōx, āter, and ātarš contain the zero grade of the root *HeHt-. - (http://institucional.us.es/revistas/habis/38/10%20zavaroni.pdf)

The specific origins of Etr. *atr-, Lat. āter/ātro-/ătrōx, Alb. votrë/votër, OIr. áith, Slav. vatra, and Av. ātarš can be then mentioned in the Etymology section. It looks like that even Baltic forms exist. According to this paper, *H3 is the standard symbol for *Hʷ. What does this mean acutally for *Hʷet-, *Hʷet˖r-?

We use lower-case h when we know which laryngeal it is, and upper-case H when we don't (h₁, h₂ and h₃, vs. H). It's inconsistent to mix an upper-case form like Hʷ with lower-case forms like hₓ: the paper you linked to seems to consistently use upper case for everything, so case doesn't mean anything in that context, and doesn't have to be followed. As for the *Hʷ notation itself, the problem is that it makes it look like it could be any laryngeal, but with rounding added, rather than a specific unknown sound that has the property of sometimes causing adjacent vowels to become rounded. I would convert your to notation to ours like this:

• hₓ = H
• Hʷ = h₃

Thus, *Hʷet- would be our h₃et-

Thank you for this introduction. So which reconstruction would you most prefer then? *h₂eh₁ter-, *h₂eh₂ter-, *h₂eHtr-, *HeHtr-, *HeHt(e)r-, *Hḕt-, *h₃et˖r- or *h₃et-?

I'm more familiar with the conventions used by our PIE entries than I am with the intricacies of laryngeal theory itself. I'll let CodeCat answer that