  We show that every five-dimensional Sasakian Lie algebra with trivial center
is $\varphi$-symmetric. Moreover starting from a particular Sasakian structure
on the Lie group $SL(2,\mathbb{R})\times\text{Aff}(\mathbb{R})$ we obtain a
family of contact metric $(k,\mu)$ structures whose Boeckx invariants assume
all values less than $-1$.
