In Example EMMS4 the matrix \begin{equation*} B= \begin{bmatrix} -2 & 1 & -2 & -4\\ 12 & 1 & 4 & 9\\ 6 & 5 & -2 & -4\\ 3 & -4 & 5 & 10 \end{bmatrix} \end{equation*} was determined to have characteristic polynomial \begin{equation*} \charpoly{B}{x}=(x-1)(x-2)^3 \end{equation*} and an eigenspace for $\lambda=2$ of \begin{equation*} \eigenspace{B}{2}=\spn{\set{\colvector{-\frac{1}{2}\\1\\-\frac{1}{2}\\1}}}\\ \end{equation*}
So the geometric multiplicity of $\lambda=2$ is $\geomult{B}{2}=1$, while the algebraic multiplicity is $\algmult{B}{2}=3$. By Theorem DMFE, the matrix $B$ is not diagonalizable.