SUNSimplify[exp] simplifies color algebraic expressions involving color matrices with implicit (SUNT) or explicit fundamental indices (SUNTF) as well as structure constants (SUND, SUNF) and Kronecker deltas (SD, SDF).
If the option Explicit is set to True (default is False), the structure constants will be rewritten in terms of traces. However, since traces with 2 or 3 color matrices are by default converted back into structure constants, you must also set the option SUNTraceEvaluate to False (default is Automatic) in order to have unevaluated color traces in the output.
Overview, SUNTrace, SUNT, SUNTF, SUNF, SUND, SUNTraceEvaluate.
SUNDelta[a, b] SUNDelta[b, c]
SUNSimplify[%]\delta ^{ab} \delta ^{bc}
\delta ^{ac}
SUNT[a] . SUNT[a]
SUNSimplify[%]T^a.T^a
C_F
SUNSimplify[SUNT[a] . SUNT[a], SUNNToCACF -> False]\frac{N^2-1}{2 N}
SUNF[a, r, s] SUNF[b, r, s]
SUNSimplify[%]f^{ars} f^{brs}
C_A \delta ^{ab}
SUNF[a, b, c] SUNF[a, b, c]
SUNSimplify[%]\left(f^{abc}\right)^2
2 C_A^2 C_F
SUNF[a, b, c] SUNF[d, b, c]
SUNSimplify[%]f^{abc} f^{dbc}
C_A \delta ^{ad}
SUNF[a, b, c] SUND[d, b, c]
SUNSimplify[%, Explicit -> True]d^{bcd} f^{abc}
0
SUND[a, b, c] SUND[a, b, c]
SUNSimplify[%, SUNNToCACF -> False] // Factor2\left(d^{abc}\right)^2
\frac{\left(1-N^2\right) \left(4-N^2\right)}{N}
SUNSimplify[SUND[a, b, c] SUND[e, b, c], SUNNToCACF -> False] // Simplify\frac{(N-2) (N+2) \delta ^{ae}}{N}
SUNSimplify[SUNF[a, b, c], Explicit -> True]f^{abc}
SUNSimplify[SUNF[a, b, c], Explicit -> True, SUNTraceEvaluate -> False]2 i \;\text{tr}\left(T^a.T^c.T^b\right)-2 i \;\text{tr}\left(T^a.T^b.T^c\right)
SUNSimplify[SUND[a, b, c], Explicit -> True]d^{abc}
SUNSimplify[SUND[a, b, c], Explicit -> True, SUNTraceEvaluate -> False]2 \;\text{tr}\left(T^a.T^b.T^c\right)+2 \;\text{tr}\left(T^a.T^c.T^b\right)
SUNF[a, b, c] SUNT[c, b, a]
SUNSimplify[%]f^{abc} T^c.T^b.T^a
-\frac{1}{2} i C_A C_F
SUNF[a, b, e] SUNF[c, d, e] + SUNF[a, b, z] SUNF[c, d, z]
SUNSimplify[%, SUNIndexNames -> {j}]f^{abe} f^{cde}+f^{abz} f^{cdz}
2 f^{abj} f^{cdj}
SUNSimplify[1 - SD[i, i]]2-C_A^2
SUNSimplify[SUNF[a, b, c] SUND[d, b, c]]0
SUNSimplify[SUNF[a, b, c] SUND[a, b, d]]0
SUNSimplify[SUNF[a, b, c] SUND[a, d, c]]0
SUNSimplify[SUND[a, b, c] SUND[d, b, c]]-\left(\left(4-C_A^2\right) \delta ^{ad} \left(C_A-2 C_F\right)\right)
SUNSimplify[SUNTrace[SUNT[i1, i2, i1, i2]], FCE -> True]-\frac{C_F}{2}