n=14$ (walls~14 and~17), $n=16$ (walls~16 and~20),
$n=17$ (walls~17 and~21), $n=18$ and $n=19$.  The specified walls are
disjoint doubling walls, and refer to the figures on p.~32 of
\cite{vinberg}.  Figures $1\beta$ and $1\gamma$ of
\cite{vinberg-kaplinskaja} display 3+12=15 pairwise disjoint doubling
walls of $P_{18}$, and figure $2\gamma$ displays $20$ pairwise
disjoint doubling walls of $P_{19}$.

Vinberg also found the Weyl chamber for the reflection subgroup of the
isometry group of the integer quadratic form
$$
-2x_0^2+x_1^2+\dots+x_n^2\;
$$
for $n\leq14$.
It turns out to be redoublable for $n=2$ (walls~1 and~3),
$n=3$ (walls~3 and~5), $n=9$ (walls~9 and~12, or~10 and~12), $n=10$ (walls~11
and~13), $n=11$ (walls~11 and~15), $n=13$ (walls~13 and~18, or~13
and~19, or~14 and~18, or~14 and~19) and $n=14$ (walls~15 and~20).  The
wall numbering refers to \cite[page 34]{vinberg}.

\begin{table}\small
\begin{center}
\tabcolsep=2.8pt
\begin{tabular}{l@{\kern8pt}rr@{\kern12pt}rr@{\kern12pt}rr@{\kern12pt}rr@{\kern12pt}rr@{\kern12pt}rr@{\kern12pt}rr}
&$a_0$&$b_0$&$a_1$&$b_1$&$a_2$&$b_2$&$a_3$&$b_3$&$a_4$&$b_4$&$a_5$&$b_5$&$a_6$&$b_6$\\
$r_{1}$&0&0&$-1$&0&1&0&0&0&0&0&0&0&0&0\\
$r_{2}$&0&0&0&0&$-1$&0&1&0&0&0&0&0&0&0\\
$r_{3}$&0&0&0&0&0&0&$-1$&0&1&0&0&0&0&0\\
$r_{4}$&0&0&0&0&0&0&0&0&$-1$&0&1&0&0&0\\
$r_{5}$&0&0&0&0&0&0&0&0&0&0&$-1$&0&1&0\\
$r_{6}$&0&0&0&0&0&0&0&0&0&0&0&0&$-1$&0\\
$r_{7}$&1&0&1&1&0&0&0&0&0&0&0&0&0&0\\
$r_{8}$&1&1&1&1&1&1&1&1&0&0&0&0&0&0\\
$r_{9}$&2&1&1&1&1&1&1&1&1&1&1&1&1&0\\
$r_{10}$&2&1&2&1&1&1&1&1&1&1&1&0&0&0\\
$r_{11}$&3&2&3&2&3&2&1&1&1&1&1&1&0&0\\
$r_{12}$&2&2&2&2&2&1&1&1&1&1&1&1&1&1\\
$r_{13}$&4&2&3&2&3&2&2&2&2&1&1&1&1&1\\
$r_{14}$&7&5&7&5&6&4&4&3&3&2&3&2&2&1\\
$r_{15}$&4&3&4&3&3&2&3&2&2&1&2&1&1&1\\
$r_{16}$&4&3&4&3&4&3&2&1&1&1&1&1&1&1\\
$r_{17}$&6&4&5&4&5&4&3&2&3&2&2&2&2&1\\
$r_{18}$&6&4&6&4&5&4&3&2&2&2&2&1&2&1\\
$r_{19}$&8&5&7&5&7&5&3&3&3&2&3&2&3&2\\
$r_{20}$&8&5&8&5&7&5&3&2&3&2&3&2&2&2\\
$r_{21}$&8&6&8&5&7&5&5&3&4&3&4&3&2&1\\
$r_{22}$&6&4&6&4&5&3&3&2&3&2&3&2&1&1\\
$r_{23}$&6&4&6&4&5&4&3&2&3&2&1&1&1&1\\
$r_{24}$&6&5&6&4&6&4&4&3&3&2&3&2&1&1\\
$r_{25}$&10&7&9&7&9&6&5&4&5&4&3&2&3&2\\
$r_{26}$&10&8&10&7&10&7&6&4&5&4&3&2&2&2\\
$r_{27}$&8&6&8&6&7&5&5&3&3&2&3&2&3&2\\
$r_{28}$&12&9&12&8&11&8&7&5&5&3&5&3&4&3\\
$r_{29}$&10&7&10&7&8&6&6&4&5&3&3&2&3&2\\
$r_{30}$&10&7&10&7&9&6&5&3&4&3&4&3&3&2\\
$r_{31}$&14&10&14&10&12&8&8&5&7&5&5&3&4&3\\
$r_{32}$&10&8&10&7&10&7&6&4&4&3&4&3&3&2\\
$r_{33}$&12&8&12&8&10&7&6&4&6&4&4&3&3&2\\
$r_{34}$&12&9&12&8&11&8&7&5&6&4&4&3&3&2\\
\end{tabular}
\end{center}
\medskip
\caption{Simple roots for Bugaenko's polyhedron in $H^6$.  Each root
  has coordinates $(a_0+b_0\sqrt2,\dots,a_6+b_6\sqrt2)$.}
\label{tab-bugaenko-roots}
\end{table}

\begin{table}\small
\begin{center}
\tabcolsep=2.5pt
\begin{tabular}{ccccc|ccccc|ccccc|ccccc|ccccc|ccccc|cccc}
$\bullet$&3& & & & &8& & &3& &4& &8&u& & &3& &3&3&u&4& &4& &u&3&u&u&u& &u&4\\
3&$\bullet$&3& & & & & & & &u&3&3&u& &u&u&u&u&u&u&u&u&u&u&u&u&u&u&u&u&u&u&u\\
 &3&$\bullet$&3& & & &8& & & & &4&8&u&4& &3&4& &3& & &u& &3&u&u&u&4&3&u& &u\\
 & &3&$\bullet$&3& & & & &4& & &3& &