Noncommutation is fundamental to quantum theory. The incompatibility of observables is one crucial difference between classical and quantum mechanics. Thermodynamic laws must apply across classical and quantum systems. Historically, researchers have implicitly assumed that conserved quantities in thermodynamic systems commute with each other. When we remove this assumption, conserved quantities that fail to commute with each other in thermodynamics engender new physics. One seminal result in thermodynamics for closed quantum many-body systems is the eigenstate thermalization hypothesis (ETH), which explains how a quantum many-body system thermalizes internally. The ETH applies across many fields including atomic, molecular, and optical physics, condensed-matter physics, and high-energy physics. Murthy et al. recently showed that the ETH does not apply to systems with noncommuting conserved quantities. Murthy et al. posited a non-Abelian ETH to account for systems with noncommuting conserved quantities. We calculate numerics to support the non-Abelian ETH. We model a one-dimensional (1D) next-nearest-neighbor Heisenberg chain of 18 qubits. We represent local operators with matrices relative to an energy eigenbasis. Our numerics evidence the non-Abelian ETH’s qualitative predictions. Noh et al. also recently derived a fluctation-dissipation theorem (FDT) from the ETH. With the recently proposed non-Abelian ETH, we begin numerical calculations in support of an FDT derived from the non- Abelian ETH. We offer the first comprehensive numerical tests of the non-Abelian ETH and initial numerics for deriving an FDT for systems that follow the NAETH.