  We introduce two uncertainty relations based on the state-dependent norm of
commutators, utilizing generalizations of the B\"ottcher-Wenzel inequality. The
first relation is mathematically proven, while the second, tighter relation is
strongly supported by numerical evidence. Both relations surpass the
conventional Robertson and Schr\"odinger bounds, particularly as the quantum
state becomes increasingly mixed. This reveals a previously undetected
complementarity of quantum uncertainty, stemming from the non-commutativity of
observables. We also compare our results with the Luo-Park uncertainty
relation, demonstrating that our bounds can outperform especially for mutually
unbiased observables.
