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Let $C$ be a smooth irreducible complex projective curve of genus $g$ and let$B^k(2,K_C)$ be the Brill-Noether loci parametrizing classes of (semi)-stablevector bundles $E$ of rank two with canonical determinant over $C$ with$h^0(C,E)\geq k$. We show that $B^4(2,K_C)$ it has an irreducible component$\mathcal B$ of dimension $3g-13$ on a general curve $C$ of genus $g\geq 8$.Moreover, we show that for the general element $[E]$ of $\mathcal B$, $E$ fitsin a exact sequence $0\to\mathcal O_C(D)\to E\to K_C(-D)\to 0$, with $D$ ageneral effective divisor of degree three, and the corresponding coboundary map$\partial: H^0(C,K_C(-D))\to H^1(C,\mathcal O_C(D))$ has cokernel of dimensionthree.

Remarks on Ruled surfaces and rank two bundles with canonical determinant and 4 sections