On the decomposition numbers of ${\mathrm{S}\mathrm{O}}_8^{+}(2^f)$

Let $q=2^f$, and let $G={\mathrm{S}\mathrm{O}}_8^{+}(q)$ and $U$ be a Sylow $2$-subgroup of $G$. We first describe the fusion of the conjugacy classes of $U$ in $G$. We then use this information to prove the unitriangularity of the $\mathrm{\ell }$-decomposition matrices of $G$ for all $\mathrm{\ell }\mathrm{\ne }2$ by inducing certain irreducible characters of $U$ to $G$; the characters of $U$ of degree $q^3\mathrm{/}2$ play here a major role. We then determine the $\mathrm{\ell }$-decomposition matrix of $G$ in the case $\mathrm{\ell }\mid q+1$, when $\mathrm{\ell }\ge 5$ and $(q+1{)}_{\mathrm{\ell }}>5$, up to two non-negative indeterminates in one column.