We prove that exterior powers of (skew-)symmetric bundles induce a $lambda$-ring structure on the ring $GW^0(X) oplus GW^2(X)$, when $X$ is a scheme where $2$ is invertible. Using this structure, we define stable Adams operations on Hermitian $K$-theory. As a byproduct of our methods, we also compute the ternary laws associated to Hermitian $K$-theory.
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