How to annotate lines in split equation with left-justified comments?

I have an equation that is split into 4 lines using amsmath's split environment. How can I annotate individual lines with a text comment such that the comments are themselves left-justified and aligned? So what I want is (without the equation number):

  lhs = some term         (comment1)
+ another term
+ yet another term  (comment2)
+ final term        (comment3)


In principle, left-justified and aligned comments can be obtained with the alignat environment, of course. However, while I can select which of the lines get numbered using \notag so I get a single equation number, the number won't be vertically centered (because I have an even number of lines).

I've searched for solutions for a while but have come up empty. The following MWE is incomplete because it does not contain the left-justified and aligned comments:

\documentclass{minimal}
\usepackage{amsmath}
\begin{document}
$$\begin{split} \alpha & = \alpha_0 \\ & + \beta^2 (H-\sqrt{f/g}) \end{split}$$
\end{document}


Thanks for any help.

You can use aligned or alignedat{2}, and take into account or not the widths of the comments:

\documentclass{article}
\usepackage[showframe]{geometry}
\usepackage{amsmath}

\begin{document}

\begin{aligned} α& = \alpha₀ & & \text{\footnotesize(a first comment)} \\ & + \beta² (H-√{f/g}) & & \text{\footnotesize(another comment)} \end{aligned}
\vskip 0.5cm
\begin{alignedat}{2} α& = \alpha₀ &\qquad & \rlap{\footnotesize(a first comment)}\\ & + \beta² (H-√{f/g}) & & \rlap{\footnotesize(another comment)} \end{alignedat}
\vskip 0.5cm
\begin{alignedat}{2} α& = \alpha₀ &\qquad & \text{\footnotesize(a first comment)}\\ & + \beta² (H-√{f/g}) & & \text{\footnotesize(another comment)} \end{alignedat}

\end{document}