The root lattice of $\mathfrak{so}(7)$ is given by the following 18 roots:

  $$
\left(\begin{array}{c}0\\0\\1\end{array}\right)
    ,
    \left(\begin{array}{c}0\\0\\-1\end{array}\right)
    ,
    \left(\begin{array}{c}1\\1\\0\end{array}\right)
    ,
    \left(\begin{array}{c}-1\\1\\0\end{array}\right)
    ,
    \left(\begin{array}{c}1\\-1\\0\end{array}\right)
    ,
    \left(\begin{array}{c}-1\\-1\\0\end{array}\right)
    ,
    \left(\begin{array}{c}1\\0\\0\end{array}\right)
    ,
    \left(\begin{array}{c}-1\\0\\0\end{array}\right)
    ,
    \left(\begin{array}{c}0\\1\\0\end{array}\right)
    ,
    \left(\begin{array}{c}0\\-1\\0\end{array}\right)
    ,
    \left(\begin{array}{c}1\\0\\1\end{array}\right)
    ,
    \left(\begin{array}{c}-1\\0\\1\end{array}\right)
    ,
    \left(\begin{array}{c}0\\1\\1\end{array}\right)
    ,
    \left(\begin{array}{c}0\\-1\\1\end{array}\right)
    ,
    \left(\begin{array}{c}1\\0\\-1\end{array}\right)
    ,
    \left(\begin{array}{c}-1\\0\\-1\end{array}\right)
    ,
    \left(\begin{array}{c}0\\1\\-1\end{array}\right)
    ,
    \left(\begin{array}{c}0\\-1\\-1\end{array}\right)
    $$

Its root lattices looks like this:

[![enter image description here][1]][1]

$\mathfrak{g}_2$ is a sub algebra of $\mathfrak{so}(7)$. So it seems naturally to ask, whether there is a relation between the two root lattices. The root lattice of $\mathfrak{g}_2$ is two-dimensional.

The roots are given by:

$$
\left(\begin{array}{c}0\\\sqrt{3}\end{array}\right)
,
\left(\begin{array}{c}0\\-\sqrt{3}\end{array}\right)
,
\left(\begin{array}{c}\frac{1}{2}\\\frac{\sqrt{3}}{2}\end{array}\right)
,
\left(\begin{array}{c}\frac{1}{2}\\-\frac{\sqrt{3}}{2}\end{array}\right)
,
\left(\begin{array}{c}-\frac{1}{2}\\\frac{\sqrt{3}}{2}\end{array}\right)
,
\left(\begin{array}{c}-\frac{1}{2}\\-\frac{\sqrt{3}}{2}\end{array}\right)
,
\left(\begin{array}{c}1\\0\end{array}\right)
,
\left(\begin{array}{c}-1\\0\end{array}\right)
,
\left(\begin{array}{c}\frac{3}{2}\\\frac{\sqrt{3}}{2}\end{array}\right)
,
\left(\begin{array}{c}\frac{3}{2}\\-\frac{\sqrt{3}}{2}\end{array}\right)
,
\left(\begin{array}{c}-\frac{3}{2}\\\frac{\sqrt{3}}{2}\end{array}\right)
,
\left(\begin{array}{c}-\frac{3}{2}\\-\frac{\sqrt{3}}{2}\end{array}\right)
    $$

and they look like:


[![enter image description here][2]][2]
Is it possible to obtain some scaled version of this root lattice from a projection of the root lattice of $\mathfrak{so}(7)$ onto a particular plane?

An experimental attempt lets me rotate the root lattice of $\mathfrak{so}(7)$ that the root structure is similar to the one of $\mathfrak{g}_2$ but the lengths of merging roots do not match:

[![enter image description here][3]][3]


  [1]: https://i.sstatic.net/TM4NT1Jj.png
  [2]: https://i.sstatic.net/A2fTxHx8.png
  [3]: https://i.sstatic.net/fzuVN2v6.png