  We describe how to implement the conformal bootstrap program in the context
of the embedding space OPE formalism introduced in previous work. To take
maximal advantage of the known properties of the scalar conformal blocks for
symmetric-traceless exchange, we construct tensorial generalizations of the
three-point and four-point scalar conformal blocks that have many nice
properties. Further, we present a special basis of tensor structures for
three-point correlation functions endowed with the remarkable simplifying
property that it does not mix under permutations of the external quasi-primary
operators. We find that in this approach, we can write the $M$-point conformal
bootstrap equations explicitly in terms of the standard position space
cross-ratios without the need to project back to position space, thus
effectively deriving all conformal bootstrap equations directly from the
embedding space. Finally, we lay out an algorithm for generating the conformal
bootstrap equations in this formalism. Collectively, the tensorial
generalizations, the new basis of tensor structures, as well as the procedure
for deriving the conformal bootstrap equations lead to four-point bootstrap
equations for quasi-primary operators in arbitrary Lorentz representations
expressed as linear combinations of the standard scalar conformal blocks for
spin-$\ell$ exchange, with finite $\ell$-independent terms. Moreover, the OPE
coefficients in these equations conveniently feature trivial symmetry
properties. The only inputs necessary are the relevant projection operators and
tensor structures, which are all fixed by group theory. To illustrate the
procedure, we present one nontrivial example involving scalars $S$ and vectors
$V$, namely $\left\langle SSSV\right\rangle$.
