  Transition metal oxides exhibit various competing phases and exotic phenomena
depending on how their reaction to the rich degeneracy of the $d$-orbital.
Large spin-orbit coupling (SOC) reduces this degeneracy in a unique way by
providing a spin-orbital-entangled ground state for 4$d$ and 5$d$ transition
metal compounds. In particular, the spin-orbital-entangled Kramers doublet,
known as the $J_{\mathbf{eff}}$=1/2 pseudospin, appears in layered iridates and
$\alpha$-RuCl$_3$, manifesting a relativistic Mott insulating phase. Such
entanglement, however, seems barely attainable in 3$d$ transition metal oxides,
where the SOC is small and the orbital angular momentum is easily quenched.
From experimental and theoretical evidence, here we report on the CuAl$_2$O$_4$
spinel as the first example of a $J_{\mathbf{eff}}$=1/2 Mott insulator in 3$d$
transition metal compounds. Based on the experimental study, including
synthesis of the cubic CuAl$_2$O$_4$ single crystal, density functional theory
and dynamical mean field theory calculations reveal that the
$J_{\mathbf{eff}}$=1/2 state survives the competition with an
orbital-momentum-quenched $S$=1/2 state. The electron-addition spectra probing
unoccupied states are well described by the $j_{\mathbf{eff}}$=1/2 hole state,
whereas electron-removal spectra have a rich multiplet structure. The fully
relativistic entity found in CuAl$_2$O$_4$ provides new insight into the
untapped regime where the spin-orbital-entangled Kramers pair coexists with
strong electron correlation.
