Here is a solution with alignat and medium-sized fractions from nccmath. I replaced all occurrences of pairs of \| … \| with a command norm that can adjust the size of the delimiters, borrowed from mathtools. The placement of the comments is adjusted through the use of rlap and \mathrlap:
\documentclass{article}
\usepackage{mathtools, nicefrac, nccmath, etoolbox}
\DeclarePairedDelimiterX\norm[1]\lVert\rVert{\ifblank{#1}{\: · \:}{#1}}
\usepackage[showframe]{geometry}
\begin{document}
\mbox{}\vskip1cm
\begin{alignat*}{2}
\frac{\Re(u'')}{\bigl(1+\norm{ u'' }^2\bigr)^{\nicefrac{1}{2}}} &= \mathrlap{\frac{\Re\Bigl(\mfrac{u}{\norm{u}^2}\Bigl(\mfrac{u^hvu^j}{\norm{u}}\Bigr)⁻¹\Bigr)}{\biggl(1+\norm[\bigg]{ \mfrac{u}{\norm{ u }^2}\Bigl(\mfrac{u^hvu^j}{\norm{u}}\Bigr)⁻¹ }^2\biggr)^{\nicefrac{1}{2}}}} \\[1ex]
&= \mathrlap{\frac{\Re(u^{1-j} v⁻¹ u^{-h})}{\norm{u}\Bigl(1+\mfrac{1}{\norm{u}^2} \norm[\big]{ u^{1-j}v⁻¹u^{-h} }^2\Bigr)^{\nicefrac{1}{2}}}} \\[1ex]
& = \frac{\Re(u^{1-j} v⁻¹ u^h)}{(1+\norm{u}^2)^{\nicefrac{1}{2}}} & \qquad & \text{\footnotesize since $ v $ is a unit quaternion \rlap{and $ j+h=1 $}} \\[1ex]
& =\frac{\Re(v)}{(1+\norm{ u }^2)^{\nicefrac{1}{2}}} & & \rlap{\parbox{0.4\linewidth}{\footnotesize since $ \Re(ab)=\Re(ba) $ and $ \Re(v⁻¹) = \Re(v) $ since $ v $ is a unit quaternion }} \\
\end{alignat*}
\end{document}
Best Answer
The following code is "full" LaTeX:
If you want, you can copy and paste the math part -- the stuff between
\[
and\]
-- into latex2png.com.