= Solution
A <Bravais lattice> is a set of points generated by integer translations along three independent <primitive lattice translation vectors>:
$$
\mathbf R=n_1\mathbf a_1+n_2\mathbf a_2+n_3\mathbf a_3,\qquad n_i\in\mathbb Z.
$$
Every <lattice point> has the same environment under <lattice translations>. A <crystal structure> consists of this <Bravais lattice> with an <atomic basis of a crystal> repeated at every <lattice point>; the set of atomic positions need not itself be a <Bravais lattice>.
The two positions in the conventional cubic cell of alpha-<tungsten> give a <body-centred cubic lattice>, with one <tungsten> atom per primitive cell and two per conventional cell. One possible set of <primitive lattice translation vectors> is
$$
\boxed{\mathbf a_1=\frac a2(-1,1,1),\quad\mathbf a_2=\frac a2(1,-1,1),\quad\mathbf a_3=\frac a2(1,1,-1).}
$$
Their primitive-cell <volume> is $a^3/2$.
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