First we give an upper bound of
c
a
t
(
E
)
\mathrm {cat}{(E)}
, the L-S category of a principal
G
G
-bundle
E
E
for a connected compact group
G
G
with a characteristic map
α
:
Σ
V
→
G
\alpha : {\Sigma }V \to G
. Assume that there is a cone-decomposition
{
F
i
|
0
≤
i
≤
m
}
\{F_{i}\,\vert \,0 \leq i\leq m\}
of
G
G
in the sense of Ganea that is compatible with multiplication. Then we have
c
a
t
(
E
)
≤
M
a
x
(
m
+
n
,
m
+
2
)
\mathrm {cat}{(E)} \leq \mathrm {Max}(m{+}n,m{+}2)
for
n
≥
1
n \geq 1
, if
α
\alpha
is compressible into
F
n
⊆
F
m
≃
G
F_{n} \subseteq F_{m}\simeq G
with trivial higher Hopf invariant
H
n
(
α
)
H_n(\alpha )
. Second, we introduce a new computable lower bound,
M
w
g
t
(
X
;
F
2
)
\mathrm {Mwgt} {(X; {\mathbb {F}_2}})
for
c
a
t
(
X
)
\mathrm {cat}({X})
. The two new estimates imply
c
a
t
(
S
p
i
n
(
9
)
)
=
M
w
g
t
(
S
p
i
n
(
9
)
;
F
2
)
=
8
>
6
=
w
g
t
(
S
p
i
n
(
9
)
;
F
2
)
\mathrm {cat}({\mathbf {Spin}{(9))}}=\mathrm {Mwgt} ({\mathbf {Spin}{(9)};{\mathbb {F}_2}}) = 8 > 6 =\mathrm {wgt}({\mathbf {Spin}{(9)};{\mathbb {F}_2}})
, where
(
w
g
t
−
;
R
)
(\mathrm {wgt}{-;R})
is a category weight due to Rudyak and Strom.