Antonyms for the phrase "keeping one eye on" could include ignoring, disregarding, neglecting, overlooking, and neglectful. Each of these antonyms describes a lack of attention or focus on a particular task or individual, rather than maintaining vigilant observation and awareness. Ignoring and disregarding imply a deliberate choice to disregard something or someone, while overlooking suggests a more unintentional failure to observe. Neglecting and neglectful describe an overall lack of attention or care, rather than a specific failure to observe. Together, these antonyms illustrate the importance of maintaining focus and attention in various contexts, from personal relationships to professional obligations.