= Solution
If $|Du|\leq L$, the eigenvalues of $a^{ij}(Du)$ lie between $(1+L^2)^{-3/2}$ and $1$, so the differentiated equations are uniformly elliptic with bounded measurable coefficients depending only on $L$. Apply the <De Giorgi-Nash-Moser theorem> to each $D_ku$. Since $|D_ku|\leq L$, for every $\theta<1$,
$$
[Du]_{C^{0,\alpha}(B_\theta)}\leq C(n,L,\theta)
$$
for some $\alpha=\alpha(n,L)\in(0,1)$. The map
$$
p\longmapsto\delta_{ij}-\frac{p_ip_j}{1+|p|^2}
$$
is smooth with bounded derivative on $|p|\leq L$. Composition therefore gives
$$
\boxed{\|b^{ij}\|_{C^{0,\alpha}(B_\theta)}\leq\beta(n,L,\theta).}
$$
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