  Let $D$ be a division ring and $K$ a subfield of $D$ which is not necessarily
contained in the center $F$ of $D$. In this paper, we study the structure of
$D$ under the condition of left algebraicity of certain subsets of $D$ over
$K$. Among results, it is proved that if $D^*$ contains a noncentral normal
subgroup which is left algebraic over $K$ of bounded degree $d$, then $[D:F]\le
d^2$. In case $K=F$, the obtained results show that if either all additive
commutators or all multiplicative commutators with respect to a noncentral
subnormal subgroup of $D^*$ are algebraic of bounded degree $d$ over $F$, then
$[D:F]\le d^2$.
