  Quantum spins of mesoscopic size are a well-studied playground for
engineering non-classical states. If the spin represents the collective state
of an ensemble of qubits, its non-classical behavior is linked to entanglement
between the qubits. In this work, we report on an experimental study of
entanglement in dysprosium's electronic spin. Its ground state, of angular
momentum $J=8$, can formally be viewed as a set of $2J$ qubits symmetric upon
exchange. To access entanglement properties, we partition the spin by optically
coupling it to an excited state $J'=J-1$, which removes a pair of qubits in a
state defined by the light polarization. Starting with the well-known W and
squeezed states, we extract the concurrence of qubit pairs, which quantifies
their non-classical character. We also directly demonstrate entanglement
between the 14- and 2-qubit subsystems via an increase in entropy upon
partition. In a complementary set of experiments, we probe decoherence of a
state prepared in the excited level $J'=J+1$ and interpret spontaneous emission
as a loss of a qubit pair in a random state. This allows us to contrast the
robustness of pairwise entanglement of the W state with the fragility of the
coherence involved in a Schr\"odinger cat state. Our findings open up the
possibility to engineer novel types of entangled atomic ensembles, in which
entanglement occurs within each atom's electronic spin as well as between
different atoms.
